Riemannian Polyhedra and Liouville-type Theorems for Harmonic maps

This paper is a study of harmonic maps from Riemannian polyhedra to (locally) non-positively curved geodesic spaces in the sense of Alexandrov. We prove Liouville-type theorems for subharmonic functions and harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau Theorem on a complete (smooth) pseudomanifolds with non-negative Ricci curvature. Then we study 2-parabolic admissible Riemannian polyhedra and prove some vanishing results on them.


Introduction
Harmonic maps between singular spaces have received considerable attention since the early 1990s. Existence of energy minimizing locally Lipschitz maps from Riemannian manifolds into Bruhat-Tits buildings and Corlette's version of Margulis's super-rigidity theorem were proved in [GS92]. In [KS93] Korevaar and Schoen constructed harmonic maps from domains in Riemannian manifolds into Hadamard spaces as a boundary value problem. The book [EF01] by Eells and Fuglede contains a description of the application of the methods of [KS93] to the study of maps between polyhedra, see also [Che95,DM08,DM10].
Our first objective in this paper is to prove Liouville-type theorems for harmonic maps. We prove the analogue of the Schoen-Yau Theorem on complete (smooth) pseudomanifolds with non-negative Ricci curvature. To this end, we generalize some Liouville-type theorems for subharmonic functions from [Yau82].
The classical Liouville theorem for functions on manifolds states that on a complete Riemannian manifold with non-negative Ricci curvature, any harmonic function bounded from one side must be a constant. In [Yau82], Yau proves that there is no non-constant, smooth, non-negative, L p , p > 1, subharmonic function on a complete Riemannian manifold. He also proves that every continuous subharmonic function defined on a complete Riemannian manifold whose local Lipschitz constant is bounded by L 1 function is also harmonic. Furthermore if the L 1 function belongs to L 2 as well, and the manifold has non-negative Ricci curvature, then the subharmonic function is constant. In the smooth setting, there are two types of assumptions that  have been studied on the Liouville property of harmonic maps. One is the finiteness of the energy and the other is the smallness of the image. For example, Schoen and Yau [SY76] proved that any non-constant harmonic map from a complete manifold of non-negative Ricci curvature to a manifold of non-positive sectional curvature has infinite energy. Hildebrandt-Jost-Widman [HJW81] (see also [Hil82,Hil85]) proved a Liouville-type theorem for harmonic maps into regular geodesic (open) balls in a complete C 3 Riemannian manifold from a simple or compact C 1 Riemannian manifold. For more references for Liouville theorem for harmonic maps and functions in both smooth and singular setting see the introduction in [KS08].
A connected locally finite n-dimensional simplicial polyhedron X is called admissible, if X is dimensionally n-homogeneous and X is locally (n − 1)chainable. It is called circuit if instead it is (n − 1)-chainable and every (n − 1)-simplex is the face of at most two n-simplex and pseudomanifold if it is admissible circuit. A polyhedron X becomes a Riemannian polyhedron when endowed with a Riemannian structure g, defined by giving on each maximal simplex s of X a Riemannian metric g (bounded measurable) equivalent to a Euclidean metric on s (see [EF01]).
There exist slightly different notions of boundedness of Ricci curvature from below on general metric spaces. See for example [Stu06,LV09,Oht07,KS01,KS03] and the references therein. In the following by Ric N,µg ≥ K we mean that (X, g, µ g ) satisfies the measure contraction property. This is the convention adopted in [Oht07,Stu06]. As this definition is somewhat technical we refer the reader to Section 3 for a precise statement.
The definition of harmonic maps from admissible Riemannian polyhedra to metric spaces is similar to the one in the smooth setting. However due to lack of smoothness some care is needed in defining the notions of energy density,the energy functional and energy minimizing maps. Precise definitions and related results can be found in Subsection 2.7.
We can state now the main results which we obtain in this direction.
Theorem 1.2. Let (X, g, µ g ) be a complete non-compact pseudomanifold. Let f be continuous, weakly subharmonic and belonging to W 1,2 loc (X), such that ∇f L 1 is finite. Then f is a harmonic function. Theorem 1.3. Let (X, g, µ g ) be a complete, smooth n-pseudomanifold. Suppose X has non-negative N -Ricci curvature, Ric N,µg , for N ≥ n. Let f be a continuous, weakly subharmonic function belonging to W 1,2 loc (X) such that both ∇f L 1 and ∇f L 2 are finite. Then f is a constant function.
Here by a smooth pseudomanifold we mean a simplexwise smooth, pseudmanifold which is smooth outside of its singular set. That situation arises when the space is a projective algebraic variety. The Fubini-Study metric of projective space induces a Bergman metric out of singular set which is Kahler. The difficulty in extending existing results lie in the lack of a differentiable structure on admissible polyhedron in general, and the loss of completeness outside the singular set even in the case of smooth pseudomanifolds. Moreover the classical notion of Laplace operator doesn't exist in the non-smooth setting. To circumvent this latter problem, influenced by the work of [Gig12], we define the Laplacian of a subharmonic function as a unique Radon measure (Theorem 4.2). As a consequence we obtain Green's formula for such functions.
The following two corollaries are important consequences of theorems 1.1 and 1.3.
Corollary 1.4. Let (X, g, µ g ) be a complete, smooth n-pseudomanifold. Suppose X has non-negative N -Ricci curvature, for N ≥ n. Suppose Y is a Riemannian manifold of non-positive curvature, and u : (X, g) → (Y, h) a continuous harmonic map belonging to W 1,2 loc (X, Y ). If u has finite energy, then it is a constant map.
Corollary 1.5. Let (X, g, µ g ) be a complete, smooth n-pseudomanifold. Suppose X has non-negative N -Ricci curvature, N ≥ n . Let Y be a simply connected, complete geodesic space of non-positive curvature and u : (X, g) → Y a continuous harmonic map with finite energy, belonging to W 1,2 loc (X, Y ). If M e(u)dµ g < ∞, then u is a constant map. Our second objective in this paper is the study of 2-parabolic admissible polyhedra. We say a connected domain Ω in an admissible Riemannian polyhedron is 2-parabolic, if for every compact sets, its relative capacity with respect to Ω is zero. Our main theorem is Theorem 1.6. Let X be 2-parabolic pseudomanifold. Let f in W 1,2 loc (X) be a continuous, weakly subharmonic function, such that ∇f L 1 and ∇f L 2 are finite. Then f is constant.
In order to prove this theorem we need to generalize some of the results in [Hol90]. This is done in Subsection 5.2. In particular we will need following propositions.
Proposition 1.7. Let (X, g) be 2-parabolic admissible Riemannian polyhedron. Suppose f in W 1,2 loc (X) is a positive, continuous superharmonic function on X. Then f is constant.
Proposition 1.8. Let X be 2-parabolic admissible Riemannian polyhedron. Let f in W 1,2 loc (X) be a harmonic function such that ∇f L 2 is finite. Then f is constant.
The proofs of the propositions above follow a similar pattern as their equivalents for Riemannian manifolds. They are based on the fact that admissible Riemannian polyhedra admits an exhaustion by regular domains, and the validity of comparison principle on admissible Riemannian polyhedra. The main new element in the proof of Theorem 1.9 is 2. Preliminaries 2.1. Riemannian polyhedra. In this subsection we have gathered standard definitions and results about Riemannian polyhedra which will be used in the rest of the manuscript.
Definition 2.1. [EF01] A countable locally finite simplicial complex K, consists of a countable set {v} of elements, called vertices, and a set {s} of finite non void subsets of vertices, called simplexes, such that • any set consisting of exactly one vertex is a simplex, • any non void subset of a simplex is a simplex, • every vertex belongs to only finitely many simplexes (the local finiteness condition).
To the simplicial complex K, we associate a metric space |K| defined as follow. The space |K| is the set of all formal finite linear combinations α = v∈K α(v)v of vertices of K such that 0 ≤ α(v) ≤ 1, v∈K α(v) = 1 and {v : α(v) > 0} is a simplex of K. |K| is made into a metric space with barycentric distance d(α, β) between two points α = α(v)v and, β = β(v)v of |K| given by the finite sum With this metric |K| is locally compact and separable.
Lemma 2.2. [EF01] Let K be a countable, locally finite simplicial complex of finite dimension n, and V a Euclidean space of dimension 2n + 1. There exists an affine Lipschitz homeomorphism f of |K| onto a closed subset of V .
Definition 2.3. [EF01] We shall use the term polyhedron to mean a connected locally compact separable Hausdorff space X for which there exists a simplicial complex K and a homeomorphism θ of |K| onto X. Any such pair T = (K, θ) is called a triangulation of X.
Definition 2.4. [EF01] A polyhedron X will be called admissible if in some (hence in any) triangulation, (i) X is dimensionally homogeneous, i.e. all maximal simplexes have the same dimension n(= dim X), or equivalently every simplex is a face of some n-simplex and (ii) X is locally (n−1)-chainable, i.e. for every connected open set U ⊂ X, the open set U \X n−2 is connected. The boundary ∂X of a polyhedron X is the union of all non maximal simplexes contained in only one maximal simplex. In this paper we always assume that (X, g) satisfies ∂X = ∅.
Definition 2.5. [EF01] By an n-circuit we mean a polyhedron X of homogeneous dimension n such that in some, (and hence any) triangulation, (a) every (n − 1)-simplex is a face of at most two n-simplexes (exactly two if ∂X = 0), and 6 ZAHRA SINAEI (b) X is (n − 1)-chainable, i.e. X\X n−2 is connected, or equivalently any two n-simplexes can be joined by a chain of contiguous (n − 1)-and n-simplexes.
Let S = S(X) denote the singular set of an n-circuit X: the complement of the set of all points of X having a neighborhood which is a topological n-manifold (possibly with boundary). S is a closed triangulable subspace of X of codimension ≥ 2, and X\S is a topological n-manifold dense in X (there is a PL stratification for such spaces). An admissible circuit is called a pseudomanifold. 1 Definition 2.6. [EF01] A Lipschitz polyhedron is a metric space X which is the image of the metric space |K| of some complex K under a Lipschitz homeomorphism θ : |K| → X. The pair (K, θ) is then called a Lipscitz triangulation (or briefly a triangulation) of the Lipschitz polyhedron X.
By a null set in a Lipschitz polyhedron X is understood a set Z ⊂ X such that Z meets every maximal simplex s (relative to some, and hence any triangulation T = (K, θ) of X) in a set whose pre-image under θ has p-dimensional Lebesgue measure 0, p = dim s.
From Lemma 2.2 follows that every Lipschitz polyhedron (X, d X ) can be mapped Lipschitz homeomorphically and (simplexwise) affinely onto a closed subset of a Euclidean space. Analogously, let X denote a locally compact separable metric space, each point of which has a neighborhood that can be bi-Lipschitz embedded as a closed subset of some Euclidean space. Then there is a proper Lipschitz homeomorphism of X onto a closed subset of some Euclidean space. For a Lipschitz circuit X with singular set S, X\S is a Lipschitz manifold.
2.1.1. Riemannian Structure on a polyhedron. A Riemannian polyhedron X = (X, g) will be defined as a Lipschitz polyhedron X endowed with a covariant bounded measurable Riemannian metric tensor g s on each maximal simplex s, satisfying the ellipticity condition 1 below.
Let T = (K, θ) be a specific (Lipschitz) triangulation of the Lipschitz polyhedron X. We shall view |K| as embedded in a Euclidean space V via an affine Lipschitz homeomorphism. Suppose, to begin with, that X has homogeneous dimension n. Choose a measurable Riemannian metric g s on the open Euclidean n-simplex θ −1 (s • ) of |K| ⊂ V . In terms of Euclidean coordinates x 1 , . . . , x n of points x = θ −1 (p) ∈ θ −1 (s • ) (in its affine span in V ), g s thus assigns to almost every point p ∈ s • , or to x ∈ θ −1 (s • ), an n × n symmetric positive definite matrix g s (x) = (g ij (x)) i,j=1...,n 1 In many texts the term pseudomanifold used for what we called a circuit. LIOUVILLE-TYPE THEOREMS   7 with measurable real entries, and there is a constant Λ s > 0 such that (with the usual summation convention) for a.e. x ∈ θ −1 (s • ) and every ξ = (ξ 1 , . . . , ξ n ) ∈ R n . This condition is independent of the choice of Euclidean frame on θ −1 (s • ). The second inequality in 1 amounts to the components of g s being bounded. If X is not necessarily dimensionally homogeneous, define g, as above for each homogeneous subpolyhedron X n , where X n denotes the union of all maximal n-simplexes of X.
The covariantly defined map g : s → g s , thus defined on the set S * (X, T ) of all maximal simplexes s of T , is called the Riemannian (polyhedral) structure, or metric, on X.
We require that, relative to a fixed triangulation T of a Riemannian polyhedron X, Λ := sup{Λ s : s ∈ S * (X, T )} < ∞. Note that Λ here depends on the choice of constants λ s . The smallest such constant Λ will be called the ellipticity constant of X = (X, T, g).
A Riemannian metric g on a polyhedron X is said to be continuous, if relative to some (hence any) triangulation, g s is continuous up to the boundary on each maximal simplex s, and for any two maximal simplexes s and s ′ sharing a face t, g s and g s ′ induce the same Riemannian metric on t. There is a similar notion of a Lipschitz continuous Riemannian metric.
Definition 2.7. [EF01] Let Z denote the collection of all null sets Z ⊂ X. For a given triangulation T = (K, θ) consider in particular the set Z T ∈ Z obtained from X by removing from each maximal simplex s in X those points of s • which are Lebesgue points for g s (that is, for every component g). For x, y ∈ X and any Z ∈ Z such that Z ⊃ Z T we denote by Lip Z (x, y, X) the family of all Lipschitz continuous paths γ : [a, b] → (X, e) with γ(a) = x, γ(b) = y which are transversal to Z in the sense that γ −1 (Z) is a null subset of [a, b]. Here e denotes the metric induced by the Euclidean distance on a Euclidean space V in which |K| is affinely Lipscitz embedded. The length L T (γ) of such a path γ is well defined by where (γ 1 , . . . , γ n ) = θ −1 • γ in terms of Euclidean coordinates on the open Euclidean simplex θ −1 (s • ), and the dot means differentiation. Write Finally set

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By the definition above, where d e X is the distance defined as above for Euclidean Riemannian metric g e on X.
Proposition 2.8. [EF01] a) The distance function d g X = d X on a Riemannian polyhedron (X, g) is intrinsic, in particular independent of the chosen triangulation. ( Here Lip(X) denotes the class of Lipschitz continuous functions u : X → R and |∇u| is the norm of the Riemannian gradient. The latter is defined a.e. in X (that is, a.e. in each s ∈ S * (X, T )), for a given triangulation T = (K, θ) of X, by Rademacher's theorem for Lipschitz functions on Euclidean domains, applied to u expressed in Euclidean coordinates in the interior of θ −1 (s). On geometrical and topological properties. Here we collect some geometrical and topological properties of Riemannian polyhedra that we need afterwards. For the Riemannian polyhedron (X, g, d X ) we have, 1. The concepts of orientation and degree of mappings make sense for circuits, also X\S induce the same orientation from X. 2. If X is circuit then it is non-branching. 3. A polyhedron is not necessarily an Alexandrov space (Alexandrov curvature, κ > −∞). For example for the pinched torus we have κ ≥ −∞. 4. We call a pseudomanifold (X, g, d X ) a Lipschitz pseudomanifold, if g is Lipschitz continuous. If g is simplexwise smooth such that (X\S, g| X\S ) has the structure of a smooth Riemannian manifold, we call (X, g, d X ) a smooth pseudomanifold.
2.2. The Sobolev space W 1,2 (X). Let (X, g, d X ) denote an admissible Riemannian polyhedron of dimension n. We denote by Lip 1,2 (X) the linear space of all Lipschitz continuous functions u : (X, d X ) → R for which the Sobolev (1, 2)-norm u defined by is finite, S (n) (X) denoting the collection of all n-simplexes s of X (relative to a given triangulation), and |∇u| the Riemannian length of the Riemannian gradient on each s. Here and elsewhere, the integration is with respect to the Riemannian volume measure on X, respectively on each s in S (n) (X), unless otherwise specified.
The Lebesgue space L 2 (X) = s∈S (n) (X) L 2 (s) is likewise formed with respect to the volume measure.
The Sobolev space W 1,2 (X) is defined as the completion of Lip 1,2 (X) with respect to the above Sobolev norm · . We use the notions Lip c (X), W 1,2 0 (X) and W 1,2 loc (X), in order for the linear space of functions in Lip(X) with compact support, the closure of Lip c (X) in W 1,2 (X) and all u ∈ L 2 loc (X) such that u ∈ W 1,2 (U ) for all relatively compact subdomains U in X. Sobolev space on metric spaces. For the definitions and theorems in this part, we refer the reader to [Che99]. Let (Y, d, µ) be a metric measure space, µ Borel regular. Assume also the measure of balls of finite and nonzero radius are finite and non-zero. Fix a set A ⊂ Y . Let f be a function on A with values in the extended real numbers.
Definition 2.9. An upper gradient, for f is an extended real valued Borel function, g : A → [0, ∞], such that for all points, y 1 , y 2 ∈ A and all continuous rectifiable curves, c : [0, l] → A, parameterized by arc length s, with c(0) = y 1 , c(l) = y 2 , we have Note that in above definition the left-hand side is interpreted as ∞, if either f (y 1 ) = ±∞ or f (y 2 ) = ±∞. If on the other hand, the right-hand side is finite then it follows that f (c(s)) is a continuous function of s. For a Lipschitz function f we define the lower pointwise Lipschitz constant of f at x as For any Lipschitz function f we have lip f (x) ≤ Lip f . In the special case We now define the Soblolev space H 1,p , for 1 ≤ p < ∞.
Definition 2.10. Whenever f ∈ L p (Y ), let where the infimum is taken over all sequence {g i }, for which there exists a sequence f i L p −→ f , such that g i is an upper gradient for f i , for all i.
For p ≥ 1, the Sobolev space, H 1,p , is the subspace of L p consisting of functions, f , for which f 1,p < ∞, equipped with the norm · 1,p . The space H 1,p is complete. Next we define the notions of generalized upper and minimal upper gradients. This will allow us to give a nice interpretation of the H 1,p norm of Sobolev functions.
Definition 2.11. The function, g ∈ L p is a generalized upper gradient for Definition 2.12. For fixed p, a minimal generalized upper gradient for f is a generalized upper gradient g f , such that f 1,p = f L p + g f L p .
Theorem 2.13. For all 1 < p < ∞ and f ∈ H 1,p there exists a minimal generalized upper gradient, g f , which is unique up to modification on subsets of measure zero.
We will discuss two important properties of metric spaces called the ball doubling property and the Poincaré inequality for functions on them. These are essential assumptions to get a richer theory on metric spaces.
Definition 2.14. Let (Y, d, µ) be a metric measure space. The measure µ is said to be doubling (locally) if for all r ′ there exists κ = κ(r ′ ) such that for all y ∈ Y and 0 < r < r ′ 0 < µ(B r (y)) ≤ 2 κ µ(B r/2 (y)). (2) Definition 2.15. Let q ≥ 1. We say that Y supports a weak Poincaré inequality of type (q, p), if for all r ′ > 0, there exist constants 1 ≤ λ < ∞ and C = C(p, r ′ ) > 0 such that for all r ≤ r ′ , and all upper gradients g of f , where f x,r := Br(x) f dµ. If λ = 1, then we say that X supports a strong (q, p)-Poincaré inequality.
By Theorem 4.24 in [Che99], for any metric space which satisfies (2) and (3), for some 1 ≤ p < ∞ and q = 1, the subspace of locally Lipschitz functions is dense in H 1,p . Also by Theorem 6.1 there, on a locally complete metric space with the mentioned properties, we have for some 1 < p < ∞ and for any f ∈ H 1,p , g f = Lip f , µ-almost everywhere. Therefore, on a Riemannian polyhedron (X, g, µ g ), for any f ∈ H 1,2 , g f (y) = |∇f (y)| for a.e. y and In the following, we always consider X = (X, g, µ g ) to be an admissible Riemannian polyhedron. Some of the concepts below are defined on metric spaces in general but for simplicity we present them only on Riemannian polyhedron and for p = 2. For more information on metric spaces we refer the reader to [BB11].

Capacities.
Definition 2.16. [BB11] The capacity of a set E ⊂ X is the number where the infimum is taken over all u ∈ W 1,2 (X) such that u ≥ 1 on E.
Definition 2.17. [BB11] Assume Ω ⊂ X is bounded. Let E ⊂ Ω. We define the variational capacity where the infimum is taken over all u ∈ W 1,2 0 (Ω) such that u ≥ 1 on E. In the above definitions the infimum can be taken only over u ≤ 1 such that it is equal 1 on a neighborhood of E. Also we write cap(E) = cap(E, X). Definition 2.20. function u ∈ W 1,2 loc (X) is said to be weakly harmonic if X ∇u, ∇ρ dµ g = 0 for every ρ ∈ Lip c (X).
Proposition 2.22. A function u ∈ W 1,2 (X) is weakly harmonic if and only if u minimizes the energy E(v) among all functions v ∈ W 1,2 (X) such that v − u ∈ W 1,2 0 (X). Theorem 2.23. (The variational Dirichlet problem) Suppose the following Poincaré inequality holds: with c depending only on the admissible Riemannian polyhedron X. For any f ∈ W 1,2 (X) the class of competing maps contains a unique weakly harmonic function u. That function is the unique solution u of the equation Corollary 2.24. Assume that Ω is bounded and such that C p (X\Ω) > 0. For any f ∈ W 1,2 (Ω), the class of functions Proof. Since X satisfies the Poincaré inequality and using Theorem 5.54 in [BB11], Ω satisfies the inequality (5). By the theorem above, there is a unique minimizer which is weakly harmonic.
Note that if X is unbounded and Omega bounded then C p (X\Ω) > 0 is provided. Using admissible Riemannian polyhedron satisfy the ball doubling property and the Poincaré inequality, Theorem 2.25. Every weakly harmonic function on X is Holder continuous (after correction on a null set).
A continuous weakly harmonic function on X is called harmonic.
Remark 1. From the discussion above one can see in the definition of variational capacity that there is a harmonic function u which takes the minimum in (4). This function is not necessarily continuous on the boundary of Ω\E.

Polar sets.
Definition 2.26. A set S ⊂ X is said to be a polar set for the capacity if for every pair of relatively compact open sets U 1 ⋐ U 2 ⊂ X such that d(U 1 , X\U 2 ) > 0 we have According to Theorem 9.52 in [BB11] (see also Section 3 in [GT02]), S is a polar set if and only if every point of X has an open neighborhood U on which there is a superharmonic function which equals ∞ at every point of S ∩ U . An equivalent formulation is to say that C 2 (S) = 0 Lemma 2.27. A closed set S ⊂ X is a polar set if and only if for every neighborhood U of S and every ǫ > 0, there exists a function Lip(X) such that i) the support of ϕ is contained in X\S; Proof. The proof is based on the definition of polar set and it is completely the same as the case of Riemannian manifolds. See Proposition 3.1 in [Tro99] for the proof of the equivalance on Riemannian manifolds.
2.6. The Dirichlet space L 1,2 0 (X). In the present setting, the Dirichlet space on the admissible Riemannian polyhedron X will be the following Hilbert space L 1,2 0 (X), a subspace of W 1,2 loc (X), and with the energy norm E(u) 1/2 . Proposition 2.28. Suppose that, for every compact set K ⊂ X, for all x ∈ Lip c (X), with c(K) depending only on X and K. In particular, X is noncompact. The completion L 1,2 0 (X) of space Lip c (X) within L 1 loc (X) with respect to the norm E(u) 1/2 is then a regular Dirichlet space of strongly local type. L 1,2 0 (X) is a subset of W 1,2 loc (X). Also, if (X, d X ) is complete then L 1,2 0 (X) is an admissible Dirichlet space.
We should note that being polar is independent of the Riemannian structure on the polyhedron. Also for local questions, condition (7) is not required (it is automatically satisfied with X replaced by the open star of a point a of X relative to a sufficiently fine triangulation and in view of inequality (3)).

2.7.
Harmonic maps on Riemannian polyhedra. The energy of maps of a Riemannian domain into an arbitrary metric space was defined and investigated by Korevaar and Schoen [KS93]. Here, we give an introduction to the concept of energy of maps, energy minimizing maps and harmonic maps on Riemannian polyherda. In the case that the target Y is a Riemannian C l -manifold the energy of a map is given by the expected explicit expression (similarly when the target is a Riemannian polyhedron with continuous Riemannian metric).
Let (X, g) be an admissible n-dimensional Riemannian Polyhedron with simplexwise smooth Riemannian metric. We do not require that g is continuous across lower dimensional simplexes. Let Y be an arbitrary metric space. Denote by L 2 loc (X, Y ) the space of all µ g -measurable maps ϕ : X → Y having separable essential range, and for which d Y (ϕ(·), q) ∈ L 2 loc (X, µ g ) for some point q (and therefore by the triangle inequality for any q ∈ Y ). For ϕ, ψ ∈ L 2 loc (X, Y ) define their distance The approximate energy density of a map ϕ ∈ L 2 loc (X, Y ) is defined for ε > 0 by The function e ε (ϕ) ≥ 0 is of class L 1 loc (X, µ g ), [KS93]. Definition 2.30. The energy E(ϕ) of a map ϕ of class L 2 loc (X, Y ) is We say that ϕ is locally of finite energy, and write ϕ ∈ W 1,2 loc (X, Y ), if E(ϕ| U ) < ∞ for every relatively compact domain U ⊂ X. For example every Lipschitz continuous map ϕ : X → Y is in W 1,2 loc (X, Y ). Now we give a necessary and sufficient condition for a map ϕ be in ϕ ∈ W 1,2 loc (X, Y ). Lemma 2.31. Let (X, g) be an admissible n-dimensional Riemannian polyhedron with simplexwise smooth Riemannian metric, and (Y, d Y ) a metric space. A map ϕ ∈ L 2 loc (X, Y ) is locally of finite energy if there is a function e(ϕ) ∈ L 1 loc (X) such that e ε (ϕ) → e(ϕ) as ε → 0, in the sense of weak convergence of measures: Energy of maps into Riemannian manifolds. Let the domain be an arbitrary admissible Riemannian polyhedron (X, g) (g is only measurable with local elliptic bounds, unless otherwise specified), and the target is a Riemannian Definition 2.32. Relative to a given countable atlas on a Riemannian C 1manifold (N, h), a map ϕ : (X, g) → (N, h) is of class W 1,2 loc (X, N ), or locally of finite energy, if (i) ϕ has a quasicontinuous version, (ii) its components ϕ 1 , . . . , ϕ m in charts η : V → R m are of class W 1,2 (U ) for every quasiopen set U ⊂ ϕ −1 (V ) of compact closure in X, and (iii) the energy density e(ϕ) of φ , defined a.e. in each of the quasiopen sets φ −1 (V ) covering X by is locally integrable over (X, µ g ).
The energy of ϕ ∈ W 1,2 loc (X, N ) is defined by E(ϕ) = X e(ϕ) dµ g . There is also corresponding definition for the energy of maps into Riemannian polyhedra. There, (X, g) is admissible, dim X = n, and g is measurable with elliptic bounds on each n-simplex of X. The polyhedron Y is not required to be admissible, but its Riemannian metric h is assumed to be continuous. Energy minimizing maps. We suppose that (X, g), n-dimensional admissible Riemannian polyhedra with g simplexwise smooth and Y any metric space.
Definition 2.33. A map φ ∈ W 1,2 loc (X, Y ) is said to be locally E-minimizing if X can be covered by relatively compact domains U ⊂ X for which E(φ| U ) ≤ E(ψ| U ) for every map ψ ∈ W 1,2 loc (X, Y ) such that φ = ψ a.e. in X\U . Some of the results concerning energy minimizing maps on Riemannian manifolds, extend to the case of Riemannian polyhedra with some restrictions on the geometry of the target.
Theorem 2.34. If Y is a simply connected complete Riemannian polyhedron of non positive curvature, every locally energy minimizing map ϕ : X → Y is Holder continuous.

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Theorem 2.35. Let X and Y be compact Riemannian polyhedra. Assume that (i) X is admissible, and (ii) Y has non positive curvature. Then every homotopy class H of continuous maps X → Y has an Eminimizer relative to H, and any such is Holder continuous.
Harmonic maps. Consider an admissible Riemannian polyhedron (X, g), dim X = n, and a metric space (Y, d Y ) Definition 2.36. A harmonic map ϕ : X → Y is a continuous map of class ϕ ∈ W 1,2 loc (X, Y ), which is bi-locally energy minimizing in the sense that X can be covered by relatively compact subdomains U , for each of which there is an open set Every continuous, locally energy minimizing map ϕ : X → Y is harmonic. Also if Y is a simply connected complete Riemannian polyhedron of nonpositive curvature, then a harmonic map ϕ : X → Y is the same as a continuous locally energy minimizing map.
Example 2.37. Let K be a compact group of isometries of smooth Riemannian manifolds (M, g), (N, h), and ψ : M → N a smooth K-equivariant map (i.e. ψ(k · x) = k · ψ(x) for all x ∈ M , k ∈ K). Then ψ induces a continuous map ϕ : X = M/K → Y = N/K. Under various circumstances, harmonicity of ψ insures that of ϕ (in the sense of above definition), and conversely.
For the definition of the energy of a map, we consider the case when (X, g) is an arbitrary admissible Riemannian polyhedron, g just bounded measurable with local elliptic bounds, dim X = n, and (N, h) a smooth Riemannian manifold without boundary, dim N = m. We denote by Γ k αβ the Christoffel symbols on N .
Definition 2.38. A weakly harmonic map ϕ : X → N is a quasicontinuous map of class W 1,2 loc (X, N ) with the following property: for any chart η : V → R n on N and any quasiopen set holds for every k = 1, . . . , m and every bounded function λ ∈ W 1,2 0 (U ).
A continuous map ϕ ∈ W 1,2 loc (X, N ) is harmonic (Definition 2.36) if and only if it is weakly harmonic (Definition 2.38).

Ricci Curvature on Riemannian Polyhedra
In the past few years, several notions of boundedness of Ricci curvature from below on general metric spaces have appeared. Sturm [Stu06] and Lott-Villani [LV09] independently introduced the so called curvature-dimension condition on a metric measure space denoted by CD(k, N ). The curvature dimension condition implies the generalized Brunn-Minkowski inequality (hence the Bishop-Gromov comparison and Bonnet-Myer's theorem ) and a Poincaré inequality (see [Stu06,LV07,LV09]). Meanwhile, Sturm and Ohta introduced a measure contraction property denoted by M CP (k, N ). The condition M CP (k, N ) also implies the Bishop-Gromov comparison, Bonnet-Myer's theorem and a Poincaré inequality (see [Stu06,Oht07]). Note that all of these generalized notions of Ricci curvature bounded below are equivalent to the classical one on smooth Riemannian manifolds. Here we define both conditions and show then that on a Riemannian polyhedron we can use both of them. In the following definitions, we always assume that (X, d) is a seprable length space , P (X) is the set of all Borel probability measures We define the non negative function p(r) = rU ′ + (r) − U (r) with p(0) = 0. Given a reference probability measure ν ∈ P 2 (X), define the function U ν : is the Lebesgue decomposition of µ with respect to ν into an absolutely continuous part ρν and a singular part µ s , and If N ∈ [1, ∞) then we define DC N to be the set of such functions U so that the function is convex on (0, ∞). We further define DC ∞ to be the set of such functions U so that the function 18 ZAHRA SINAEI is convex on (−∞, ∞). A relevant example of an element of DC N is given by Definition 3.1. Given N ∈ [1, ∞], we say that a compact measured length where λ : Note that inequalities (11) and (12) are only assumed to hold along some Wasserstein geodesic from µ 0 to µ 1 , and not necessarily along all such geodesics. This is what is called weak displacement convexity.
Proposition 3.2. If a compact measured length space (X, d, ν) has non negative N -Ricci curvature for some N ∈ [1, ∞), then for all x ∈ supp(ν) and all 0 < r 1 ≤ r 2 To generalize the notion of N -Ricci curvature to the non-compact case, we always consider a complete pointed locally compact metric measure space (X, ⋆, ν). Also for U ν to be a well-defined functional on P 2 (X), we impose the We define M −∞ , by the condition X e −cd(x,⋆) 2 dν(x) < ∞, where c is a fixed positive constant. We should mention that most of the results for compact case (for example the Bishop-Gromov comparison) are valid for the noncompact case.
Measure Contraction Property M CP (K, N ). [Oht07] Let (X, d X ) be a length space, and µ a Borel measure on X such that 0 < µ(B(x, r)) < ∞ for every x ∈ X and r > 0, where B(x, r) denotes the open ball with center x ∈ X and radius r > 0.
Let Γ be the set of minimal geodesics, γ : [0, 1] → X, and define the evaluation map e t by e t (γ) := γ(t) for each t ∈ [0, 1]. We regard Γ as a subset of the set of Lipschitz maps Lip([0, 1], X) with the uniform topology. A dynamical transference plan Π is a Borel probability measure on Γ, and a path {µ t } t∈[0,1] ⊂ P 2 (X) given by µ t = (e t ) * Π is called a displacement interpolation associated to Π. For K ∈ R, we define the function holds as measures on X, where we set 0/0 = 1 and, by convention, we read Here we state two results that we are going to use in the sequel.
Proposition 3.4. Let (M, g) be an n-dimensional, complete Riemannian manifold without boundary with n ≥ 2. Then a metric measure space (M, d g , ν g ) satisfies the (K, n)-MCP if and only if Ric g ≥ K holds.
Proposition 3.5. (Bishop-Gromov volume comparison theorem) Let (X, µ) be a metric space satisfying the (K, N )-MCP. Then, for any x ∈ X, the function . According to Proposition 2.8, a Riemannian polyhedrapolyhedron (X, g, µ g ) with the metric d X = d g X is a length space. Also for any x, y ∈ X we have e(x, y) ≤ d e X (x, y). It is easy then to show that µ g is in M −2(N −1) and so on a complete Riemannian polyhedrapolyhedron we can use the notion of CD(K, N ). Also µ g is Borel and by Lemma 4.4 in [EF01], for any r there exist a constant c(r) such that is also applicable here, for N ≥ n. (By Theorem 2.4.3 in [AT04], we have the Hausdorff dimension is n and by Corollary 2.7 in [Oht07] N should be greater than n.) In the rest of this paper by Ric N,µg ≥ K we mean that (X, g, µ g ) satisfies the M CP (K, N ). In the following Lemma we show that any complete Riemannian polyhedron with non-negative Ricci curvature has infinite volume.
By Theorem 3.2, and since X is a complete locally compact length space, the above theorem is still valid for the case when X satisfies the non-negative N -Ricci curvature condition CD(0, N ), for N ∈ (1, ∞).
Remark 3. By Remark 5.8 in [Stu06] if (X, d, µ) satisfy M CP (k, N ) so does any convex set A ⊂ X. When X is a pseudomanifold, for any point x ∈ X\S, there exist a closed totally convex neighborhood V around x (for every point in a Riemannian manifold there is a geodesic ball which is totally convex). Therefore if X satisfies Ric N,µg ≥ K, so does X\S.

Some Function Theoretic Properties On Complete
Riemannain Polyhedra 4.1. Liouville-type Theorems for Functions. The aim of this subsection is to generalize some of the results in [Yau82] in order to prove some vanishing theorems for harmonic maps on Riemannian polyhedra. In [Yau82], Yau used the Gaffney's Stokes theorem on complete Riemannian manifolds to prove that every smooth subharmonic function with bounded ∇f L 1 is harmonic. He uses this fact to prove there is no non-constant L p , p > 1, non-negative subharmonic function on a complete manifold. We will prove this theorem on admissible polyhedra for p = 2. There we do not need more than W 1,2 loc -regularity for the function f . Theorem 4.1. Suppose (X, g) is a complete, admissible Riemannian polyhedron, and f ∈ W 1,2 loc (X) ∩ L 2 (X) is a non-negative, weakly subharmonic function. Then f is constant.

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Since the last line is a quadratic form which is always non-negative, we have Sending R to infinity and using the fact that f has finite L 2 -norm, we conclude that Since X is admissible, f is constant on X. (First we prove that f is constant on each maximal n-simplex S and then using the n − 1-chainability of X, we prove this in the star of any vertex p of X and then by connectedness on X.) In the following theorem, we show that the Laplacian of every weakly subharmonic function f ∈ W 1,2 loc (X) on a pseudomanifold in the distributional sense is a locally finite Borel measure. This gives us a verifying of Green's formula on these spaces. We then use this theorem, to prove that a continuous weakly subharmonic function with finite ∇f L 1 on a complete normal circuit is harmonic. Since we do not have Bochner formula on circuits, we are not able to prove the equivalent of Proposition 2 in [Yau82] in general, but only in the special case of smooth pseudomanifolds.
Theorem 4.2. Let (X, g, µ g ) be a n-pseudomanifold. Let f be a weakly subharmonic function in W 1,2 loc (X), such that ∇f L 1 is finite. Then there exist a unique locally finite Borel measure m f such that Proof. We consider the Lipschitz manifold M = X\S and the chart is a distribution on D α = Lip c (ψ α (U α )) (with respect to the topology of uniform convergence on compact sets). The linearity is obvious. We have and so Λ α is continuous. Actually one can easily show that the system of {Λ α } determines a distribution on M . f is subharmonic and so Λ α is a unique positive Radon measure (this comes from the fact that Lip c (X) is dense in C c (X)). It follows that there is a positive Radon measure m α such that Obviously m f is positive and locally finite. The uniqueness comes from the uniqueness of m α .
Gigli introduced the notion of Laplacian as a set of locally finite Borel measure (see Definition 4.4 in [Gig12]). There he proved that on infinitesimally Hilbertian spaces this set contains only one element 2 . Admissible Riemannian polyhedra are the examples of infintesimally Hilbertian space.
Theorem 4.3. Let (X, g, µ g ) be a complete non-compact n-pseudomanifold. Let f be a continuous weakly subharmonic belonging to W 1,2 loc (X) such that A 1 = ∇f L 1 is finite. Then f is a harmonic function.
Proof. To do so we consider a sequence of cut-off functions ρ n for fixed q ∈ X and R such that ρ n is equal to 1 on B(q, R) and its support is in B(q, R + n). f is a subharmonic function which satisfies the condition of lemma above, so there is a unique Borel measure m f such that Let h be any function in Lip c (X) with support in B(q, R). We have and tending n to infinity, we have and implying that f is harmonic.
Now we prove the equivalent of Proposition 2 in [Yau82]. We give here another proof of the theorem above for smooth pseudomanifolds under the extra assumption that f should have finite energy. Instead of Theorem 4.2, we goal Cheeger's Green formula on compact smooth pseudomanifolds in the proof.

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Theorem 4.4. Let (X, g, µ g ) be a complete non-compact smooth pseudomanifold. Suppose X has non-negative N -Ricci curvature, Ric N,µg , for N ≥ n. Let f be a continuous weakly subharmonic function belonging to W 1,2 loc (X), such that both A 1 = ∇f L 1 and A 2 = ∇f L 2 are finite. Then f is constant.
Proof. We present the proof in several steps.
Step 1. We consider a sequence of cut-off functions ρ n as above such that the support of ρ n is in B(q, R + n) for fixed q ∈ X\X n−2 and some R and ρ n is equal to 1 on B(q, R).
Step 2. The (n − 2)-skeleton in X, X n−2 , is a polar set. We consider, By the definition of polar set, for every couple of open sets like U j and U j−1 , we have cap(X n−2 ∩ U j , U j−1 ) = 0. This means that for every ǫ j = 1 j , there exists a function ϕ j ∈ Lip(X) such that ϕ j ≡ 1 in a neighborhood of X n−2 ∩ U j and ρ j is zero outside U j−1 and X |∇ϕ j | 2 < 1 j . Moreover we have 0 ≤ ϕ j ≤ 1. The function η j = 1 − ϕ j has this property that the closure of its support, supp η j , is contained in X\X n−2 and the set K j = supp η j ∩ B(q, R + j), is a compact set. Furthermore K j s make an exhaustion of M = X\X n−2 .
Step 3. According to Theorem 2 in [GW79], on each K j , there exist a smooth subharmonic function Step 4. Let (Y, h) be a closed n-dimensional admissible Riemannian polyhedron, then for ζ, ψ ∈ Dom(∆) we have the following Stokes theorem on As in [Che80], every closed smooth pseudomanifold (Y, h) such that h is equivalent to some piecewise flat metric is admissible (in the sense of Cheeger). Now suppose Y j is an arbitrary Riemannian polyhedron containing B(q, R+ j). We consider its double Y j and equip it with a Riemannian metric g j , coming from the induced metric on Y j on both copies. With this metric, Y j is an admissible closed Riemannian pseudomanifold. (because the metric g j on Y j is equivalent to piecewise flat metric g e as it is constructed in section 2.1).
We extend ρ j to Y j such that it is zero on the copy of Y j and f j and η j such that they are the same functions on the copy of Y j . The function f j is in W 1,2 loc (Y j ). (see Theorem 1.12.3. in [KS93]).
By applying formula (16) on Y j , for the functions f j and ξ j = ρ j · η j , we obtain Step 5. In this step, we prove that f is harmonic on M . By use of this fact Let h be any smooth function with compact support in M ∩ B(q, R). Then there is a K m such that the support of h is in B(q, R) ∩ K m . For j large enough we will have ξ j ≡ 1 on K m and so we have considering the formula (16) as above, for j large enough we have Letting j go to infinity, we got M ∆h · f dµ g = 0. By use of Weyl's lemma f is a smooth harmonic function.
Step 6. Now we show f is constant. Since M has non-negative Ricci curvature, by the Bochner formula |∇f | is subharmonic on M and so on X. By Lemma 1.1, |∇f | is constant. Since the L 2 -norm of |∇f | is finite we have |∇f | ≡ 0. By Lemma 3.6, f should be constant.

Vanishing Results for Harmonic Maps on Complete Smooth
PseudoManifolds.
Proposition 4.5. Let (X, g, µ g ) be a complete n-dimensional Riemannian pseudomanifold. Suppose X has non-negative N -Ricci curvature, Ric N,µg , for N finite. Suppose Y is a Riemannian manifold of non-positive curvature and u : (X, g) → (Y, h) a continuous harmonic map belonging to W 1,2 loc (X, Y ). If u has finite energy, then it is a constant map.
Proof. Let u : (X, g) → (Y, h) be a continuous harmonic map in W 1,2 (X, Y ). By Remark 3, we know that on the Riemannian manifold M = X\S we have non-negative Ricci curvature.
We show that for ǫ > 0, e(u) + ǫ is weakly subharmonic on X. As the restriction maps u = u| M : (M, g) → Y is harmonic, we have a Bochner type formula for harmonic map on M and and so for ǫ > 0, on X\S We have e(u) + ǫ in W 1,2 loc (X, Y ) is weakly subharmonic on X: As in the proof of Theorem 1.1, (note that e(u) + ǫ satisfies all the assumptions of the Theorem 1.1, except the finiteness of L 2 -norm which we do not need in this step).
Letting ǫ → 0 gives and letting R → ∞ and by finiteness of the energy we have which implies that e(u) is constant. If e(u) is not zero everywhere this means that the volume of X is finite. By Lemma 3.6, this is impossible and so u is constant.
Now we extend this theorem to the case when the target Y is a metric space. By the following lemma we have the function d(u(·), q), where q is an arbitrary point in Y , is subharmonic under suitable assumption on the curvature of Y . We refer the reader to [EF01] Lemma 10.2, for the proof.
Lemma 4.6. Let (X, g) be an admissible Riemannian polyhedron, g simplexwise smooth. Let (Y, d Y ) be a simply connected complete geodesic space of non-positive curvature, and let u ∈ W 1,2 loc (X, Y ) be a locally energy minimizing map. Then u is a locally essentially bounded map and for any q ∈ Y , the function √ u q = d(u(·), q) of class W 1,2 loc (X, Y ) is weakly subharmonic, and in particular essentially locally bounded. Hence u q ∈ W 1,2 loc (X, Y ) and u q is itself weakly subharmonic.

Now we have
Proposition 4.7. Let (X, g, µ g ) be a complete non-compact smooth pseudomanifold. Suppose X has non-negative N -Ricci curvature, Ric N,µg , for N finite. Let Y be a simply connected complete geodesic space of non-positive curvature and u : (X, g) → Y a continuous harmonic map with finite energy belonging to W 1,2 loc (X, Y ). If M e(u)dµ g < ∞, then u is a constant map. Proof. According the lemma above the function v(x) = d(u(x), u(x 0 )) for some x 0 ∈ X, is weakly subharmonic. We know that |∇v| 2 ≤ ce(u), where c is a constant. v is a continuous subharmonic function whose gradient is bounded by an L 1 and L 2 integrable function. According to Lemma 1.1, v is a constant function and so u is a constant map.

2-Parabolic Riemannian Polyhedra
5.1. Definition and Properties. In this subsection we define the notion of 2-parabolicity for admissible Riemannian polyhedra. This is just the generalization of the concept of 2-parabolicity on Riemannian manifold and it coincides with the one in [GT02] (Definition 2.11) for metric measure space in general. We then prove some classical results on these spaces. We should mention that all the results which will appear in this subsection and the next subsection, except Proposition 5.5 and its corollaries are the generalizations of equivalent ones for Riemannian manifolds and the proofs are exactly the same.
Definition 5.1. Let (X, g) be an admissible Riemannian polyhedron. Let Ω be a connected domain in X. We say Ω is 2-parabolic if for every compact set D ⊂ X with non-empty interior, cap(D, Ω) = 0. Otherwise we call (X, g) 2-hyperbolic.
Proof. First suppose Ω is 2-parabolic. Then every compact set D ⊂ Ω, with nonempty interior satisfies cap(D, Ω) = 0. We choose an exhaustion D ⊂ D 1 ⊂ D 2 ⊂ . . . ⊂ Ω of Ω by compact subsets such that cap(D j , Ω) = 0 for all j. Hence we can find the function ρ j ∈ Lip c (Ω) (using the fact that Lip c (Ω) is dense in W 1,2 0 (Ω)) such that ρ j ≡ 1 on D j and Ω |∇ρ j | 2 dµ g ≤ 1/ j 2 . We have constructed the desired sequence ρ j . Conversely, suppose there exist, a sequence ρ j ∈ Lip c (Ω) with the stated properties. Then we can find a compact subset B ⊂ Ω and j 0 such that ρ j ≥ 1/2 for every j ≥ j 0 . It follows that cap(B, Ω) = 0 Lemma 5.3. If X is a 2-parabolic admissible Riemannian polyhedron and S ⊂ X is a polar set, then Ω := X\S is 2-parabolic.
Proof. X is 2-parabolic, so by Lemma 5.2, there are an exhaustion of X and a sequence of function ρ j ∈ Lip c (X) such that 0 ≤ ρ j ≤ 1 and ρ j → 1 uniformly on each compact set, and X |∇ρ j | 2 dµ g → 0. Also by Lemma 2.27 there exist another sequence of functions ϕ j with support in X\S such that ϕ j → 1 on each compact set of X\S and X |∇ϕ j | 2 dµ g → 0. The functions ρ j ϕ j on Ω provide the condition for 2-parabolicity in Lemma 5.2. 5.2. Some Function Theoretic Properties and Application to Harmonic Maps. In this subsection we prove the analogue of Theorem 1.3, for 2-parabolic admissible Riemannian polyhedra.
Proposition 5.4. Let (X, g) be 2-parabolic admissible Riemannian polyhedron. Suppose f in W 1,2 loc (X) is a positive, continuous superharmonic function on X. Then f is constant.
Proof. Since f is continuous, for any ǫ and at any point x 0 in X there exist a relatively compact neighborhood B 0 of x 0 such that f (x) > f (x 0 ) − ǫ on B 0 (we choose the triangulation such that x 0 is a vertex and the star of x 0 is equal to B 0 ). X is 2-parabolic, so cap(B 0 , X) = 0. Consider an exhaustion of X by regular domains U i such that B 0 ⋐ U 1 ⋐ U 2 ⋐ . . . ⋐ X. By Corollary 11.25 in [BB11], such exhaustion exists. There exist functions u i which are harmonic on U i \B 0 , u i ≡ 1 on B 0 and u i ≡ 0 on X\U i (see Lemma 11.17 and 11.19 in [BB11].) Also maximum principle (see Theorem 5.3 [EF01] or Lemma 10.2 [BB11] for the comparison principle). implies Define the function h i = (f (x 0 )− ǫ)u i . X is 2-parabolic, and so lim i→∞ h i = u(x 0 ) − ǫ. On the other hand f ≥ h i on the boundary of U i \B 0 . By the . By the same argument we obtain f > f (x 1 ). This is a contradiction and thus f is constant.
Proposition 5.5. Let X be 2-parabolic pseudomanifold. Let f in W 1,2 loc (X) be a continuous weakly subharmonic function such that ∇f L 1 and ∇f L 2 are finite. Then f is harmonic.
Proof. X is 2-parabolic so by Lemma 5.2, there is a sequence ρ j which converges to 1 uniformly on compact sets. Suppose we construct D, D j and ρ j as in the proof of lemma 5.2.
By Lemma 4.2, there is a locally finite Borel measure m f such that

Now let h be an arbitrary test function in Lip
and so f is harmonic on X.
Proposition 5.6. Let X be 2-parabolic admissible Riemannian polyhedron. Let f in W 1,2 loc (X) be a harmonic function such that ∇f L 2 is finite. Then f is constant function.
Proof. Set Let U i be an exhaustion of X by regular domains U i ⊂ U i+1 ⋐ X. There is function u i,j such that u i,j are harmonic on U j and u i,j = f i in X\U i . Also u i,j is continuous on X and ∇u i,j L 2 is finite. We have −i ≤ u i,j ≤ i . According to Theorem 6.3 in [EF01], u i,j are locally uniformly holder continuous, and so there is a subsequence which converges locally uniformly to some u i . The function u i is bounded and harmonic and hence is constant. Moreover u i,j − f i ∈ L 1,2 0 and so f i ∈ L 1,2 0 . Therefore and f is constant.

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Corollary 5.7. Let (X, g) be a 2-parabolic pseudomanifold with g simplexwise smooth. Let Y be a simply connected, complete, geodesic space of nonpositive curvature and u : (X, g) → Y a continuous, harmonic map with finite energy, belonging to W 1,2 loc (X, Y ). If we have X e(u)dµ g < ∞ then u is a constant map.
Corollary 5.8. Let (X, g) be a 2-parabolic admissible Riemannian polyhedron with simplexwise smooth metric g. Let Y be a complete, geodesic space of non-positive curvature and u : (X, g) → Y a continuous, harmonic map belonging to W 1,2 loc (X, Y ) with bounded image. Then u is a constant map.

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Abstract. This paper is a study of harmonic maps from Riemannian polyhedra to (locally) non-positively curved geodesic spaces in the sense of Alexandrov. We prove Liouville-type theorems for subharmonic functions and harmonic maps under two different assumptions on the source space. First we prove the analogue of the Schoen-Yau Theorem on a complete (smooth) pseudomanifolds with non-negative Ricci curvature. Then we study 2-parabolic admissible Riemannian polyhedra and prove some vanishing results on them.

Introduction
Harmonic maps between singular spaces have received considerable attention since the early 1990s. Existence of energy minimizing locally Lipschitz maps from Riemannian manifolds into Bruhat-Tits buildings and Corlette's version of Margulis's super-rigidity theorem were proved in [GS92]. In [KS93] Korevaar and Schoen constructed harmonic maps from domains in Riemannian manifolds into Hadamard spaces as a boundary value problem. The book [EF01] by Eells and Fuglede contains a description of the application of the methods of [KS93] to the study of maps between polyhedra, see also [Che95,DM08,DM10].
Our first objective in this paper is to prove Liouville-type theorems for harmonic maps. We prove the analogue of the Schoen-Yau Theorem on complete (smooth) pseudomanifolds with non-negative Ricci curvature. To this end, we generalize some Liouville-type theorems for subharmonic functions from [Yau82].
The classical Liouville theorem for functions on manifolds states that on a complete Riemannian manifold with non-negative Ricci curvature, any harmonic function bounded from one side must be a constant. In [Yau82], Yau proves that there is no non-constant, smooth, non-negative, L p , p > 1, subharmonic function on a complete Riemannian manifold. He also proves that every continuous subharmonic function defined on a complete Riemannian manifold whose local Lipschitz constant is bounded by L 1 -function is also harmonic. Furthermore if the L 1 -function belongs to L 2 as well, and the manifold has non-negative Ricci curvature, then the subharmonic function is constant. In the smooth setting, there are two types of assumptions that have been studied on the Liouville property of harmonic maps. One is the finiteness of the energy and the other is the smallness of the image. For example, as we mentioned above Schoen and Yau proved that any non-constant harmonic map from a complete manifold of non-negative Ricci curvature to a manifold of non-positive sectional curvature has infinite energy. Hildebrandt-Jost-Widman [HJW81] (see also [Hil82,Hil85]) proved a Liouville-type theorem for harmonic maps into regular geodesic (open) balls in a complete C 3 -Riemannian manifold from a simple or compact C 1 -Riemannian manifold. For more references for Liouville-type theorems for harmonic maps and functions in both smooth and singular setting see the introduction in [KS08].
A connected locally finite n-dimensional simplicial polyhedron X is called admissible, if X is dimensionally n-homogeneous and X is locally (n − 1)-chainable. It is called circuit if instead it is (n − 1)-chainable and every (n − 1)-simplex is the face of at most two n-simplex and pseudomanifold if it is admissible circuit. A polyhedron X becomes a Riemannian polyhedron when endowed with a Riemannian structure g, defined by giving on each maximal simplex s of X a Riemannian metric g (bounded measurable) equivalent to a Euclidean metric on s (see [EF01]).
There exist slightly various notions of boundedness of Ricci curvature from below on general metric spaces. See for example [Stu06, LV09, Oht07, KS01, KS03, AGS11b, EKS13, AMS13] and the references therein. In what follows by Ric N,µg ≥ K we mean that (X, g, µ g ) satisfies the either (K,N)-measure contraction property or CD(K, N ) unless otherwise specified. As these definitions are somewhat technical we refer the reader to Section 3 for a precise statement.
The definition of harmonic maps from admissible Riemannian polyhedra to metric spaces is similar to the one in the smooth setting. However due to lack of smoothness some care is needed in defining the notions of energy density, the energy functional and energy minimizing maps. Precise definitions and related results can be found in Subsection 2.4.
We can state now the main results which we obtain in this direction.
Theorem 1.1. Suppose (X, g) is a complete, admissible Riemannian polyhedron, and f ∈ W 1,2 loc (X) ∩ L 2 (X) is a non-negative, weakly subharmonic function. Then f is constant. Theorem 1.2. Let (X, g, µ g ) be a complete non-compact pseudomanifold. Let f be continuous, weakly subharmonic and belonging to W 1,2 loc (X), such that ∇f L 1 is finite. Then f is a harmonic function. Theorem 1.3. Let (X, g, µ g ) be a complete, smooth n-pseudomanifold. Suppose X satisfies CD(0, n). Let f be a continuous, weakly subharmonic function belonging to W 1,2 loc (X) such that both ∇f L 1 and ∇f L 2 are finite. Then f is a constant function.
Here by a smooth pseudomanifold we mean a simplexwise smooth, pseudomanifold which is smooth outside of its singular set. That situation arises for instance when the space is a projective algebraic variety. The difficulty in extending existing results lies in the lack of a differentiable structure on admissible polyhedron in general, and the loss of completeness outside the singular set even in the case of smooth pseudomanifolds. Moreover the classical notion of Laplace operator doesn't exist in the non-smooth setting. To circumvent this latter problem, and following the work of [Gig12], we define the Laplacian of a subharmonic function as a measure for which the Green formula holds, see Theorem 4.1.
The following two theorems are corollaries of theorems 1.1 and 1.3.
Theorem 1.4. Let (X, g, µ g ) be a complete, smooth n-pseudomanifold. Suppose X has non-negative n-Ricci curvature. Suppose Y is a Riemannian manifold of non-positive curvature, and u : (X, g) → (Y, h) a continuous harmonic map belonging to W 1,2 loc (X, Y ). If u has finite energy, and e(u) is locally bounded, then it is a constant map.
Theorem 1.5. Let (X, g, µ g ) be a complete, smooth n-pseudomanifold. Suppose X has non-negative n-Ricci curvature CD(0, n). Let Y be a simply connected, complete geodesic space of non-positive curvature and u : (X, g) → Y a continuous harmonic map with finite energy, belonging to W 1,2 loc (X, Y ). If M e(u) dµ g < ∞, then u is a constant map. Our second objective in this paper is the study of 2-parabolic admissible polyhedra. We say a connected domain Ω in an admissible Riemannian polyhedron is 2-parabolic, if for every compact set in Ω, its relative capacity with respect to Ω is zero. Our main theorem is Theorem 1.6. Let X be 2-parabolic pseudomanifold. Let f in W 1,2 loc (X) be a continuous, weakly subharmonic function, such that ∇f L 1 and ∇f L 2 are finite. Then f is constant.
Just as in the case of complete pseudomanifolds Theorem 1.7. Let (X, g) be a 2-parabolic pseudomanifold with g simplexwise smooth.
Let Y be a simply connected complete geodesic space of non-positive curvature and u : (X, g) → Y a continuous harmonic map with finite energy belonging to W 1,2 loc (X, Y ). If we have X e(u)dµ g < ∞ then u is a constant map.
Also we will obtain Theorem 1.8. Let (X, g) be a 2-parabolic admissible Riemannian polyhedron with simplexwise smooth metric g. Let Y be a complete geodesic space of non-positive curvature and u : (X, g) → Y a continuous harmonic map belonging to W 1,2 loc (X, Y ), with bounded image. Then u is a constant map.
In order to prove Theorem 1.6, we need to generalize some of the results in [Hol90]. This is done in Section 5. In particular we will need following propositions. Proposition 1.9. Let (X, g) be 2-parabolic admissible Riemannian polyhedron. Suppose f in W 1,2 loc (X) is a positive, continuous superharmonic function on X. Then f is constant. Proposition 1.10. Let X be 2-parabolic admissible Riemannian polyhedron. Let f in W 1,2 loc (X) be a harmonic function such that ∇f L 2 is finite. Then f is constant. The proofs of the propositions above follow a similar pattern as their equivalents for Riemannian manifolds. They are based on the fact that admissible Riemannian polyhedra admits an exhaustion by regular domains, and the validity of comparison principle on admissible Riemannian polyhedra. The main new ingredient in the proof of Theorem 1.6 is Proposition 1.11. Let X be a 2-parabolic pseudomanifold. Let f in W 1,2 loc (X) be a continuous, weakly subharmonic function, such that ∇f L 1 and ∇f L 2 are finite. Then f is harmonic.
The rest of this paper is organized as follows. In Section 2, we give a complete background on Riemannian polyhedron and analysis on them. Most definition and results have been taken directly from [EF01]. In Subsection 2.2, we compare the L 2 based Sobolev space on admissible Riemannian polyhedra as in [EF01], with the one in [Che99], and show that they are equal. As we couldn't find references in the literature we provide a rather detailed explanation of this fact. In Section 3, we discuss the definition of two notions of Ricci curvature, the curvature dimension condition CD(K, N ) and the measure contraction property M CP (K, N ) on metric measure spaces. We show that both notions are applicable to Riemannian polyhedra. In Proposition 3.6 we show that any non-compact complete ndimensional Riemannian polyhedron of non-negative Ricci curvature has infinite volume. Subsection 4.1 is devoted to Theorems 1.1, 1.2, 1.3 and Subsection 4.2 to Theorems 1.4, and 1.5. In Section 5 we show that as in the smooth case the "approximation by unity" property holds on admissible 2-parabolic polyhedra (Lemma 5.1). Moreover, we prove that removing the singular set of a 2-parabolic pseudomanifold yields a 2-parabolic manifold (Lemma 1.6). The rest of this Section is the detailed proof of Theorem 1.6 and its corollaries.

Acknowledgement
The Original version of this work was part of my Ph.D. dissertation. I thank my advisor Professor Marc Troyanov for his guidance and support. I also thank my Ph.D. exam jury members Professors Buser, Wenger and particularly Professor Naber who suggested an additional bound on |∇f | as hypothesis in Theorem 1.3, and so automatically on e(u) in Theorems 1.4 and 1.5. I am very grateful to thank Professor Goufang Wei, who I met as a postdoc in MSRI, for suggesting that I read papers by [HKX13,Jia13,Kel13], which allowed me to remove this assumption in Theorems 1.3, 1.5.

Preliminaries
2.1. Riemannian polyhedra. In this subsection we recall the definitions and results about Riemannian polyhedra which will be used in the rest of the manuscript. We refer the reader to the book [EF01], for more complete discussion on the subject. Simplicial complex. A countable locally finite simplicial complex K, consists of a countable set {v} of elements, called vertices, and a set {s} of finite non-void subsets of vertices, called simplexes, such that • any set consisting of exactly one vertex is a simplex.
• any non-void subset of a simplex is a simplex.
• every vertex belongs to only finitely many simplexes (the local finiteness condition).
To the simplicial complex K, we associate a metric space |K| defined as follows. The space |K| of K is the set of all formal finite linear combinations α = v∈K α(v)v of vertices of K such that 0 ≤ α(v) ≤ 1, v∈K α(v) = 1 and {v : α(v) > 0} is a simplex of K. |K| is made into a metric space with barycentric distance ρ(α, β) between two points α = α(v)v and, β = β(v)v of |K| given by the finite sum With this metric |K| is locally compact and separable. The metric ρ is not intrinsic. We denote by ρ(α, β) the length metric associated to ρ by the standard procedure [BBI01].
Lemma 2.1. [EF01] Let K be a countable, locally finite simplicial complex of finite dimension n, and V a Euclidean space of dimension 2n + 1. There exists an affine Lipschitz homeomorphism i of |K| onto a closed subset of V .
We shall use the term polyhedron to mean a connected locally compact separable Hausdorff space X for which there exists a simplicial complex K and a homeomorphism θ of |K| onto X. Any such pair T = (K, θ) is called a triangulation of X.
A Lipschitz polyhedron is a metric space X which is the image of the metric space |K| of some complex K under a Lipschitz homeomorphism θ : |K| → X. The pair (K, θ) is then called a Lipscitz triangulation (or briefly a triangulation) of the Lipschitz polyhedron X. A null set in a Lipschitz polyhedron X is understood a set Z ⊂ X such that Z meets every maximal simplex s (relative to some, and hence any triangulation T = (K, θ) of X)) in a set whose pre-image under θ has p-dimensional Lebesgue measure 0, p = dim s.
From Lemma 2.1 follows that every Lipschitz polyhedron (X, d X ) can be mapped Lipschitz homeomorphically and (simplexwise) affinely onto a closed subset of a Euclidean space.
Riemannian Structure on a polyhedron. The class of domains that we consider for our harmonic maps are Riemannian polyhedra. A Riemannian polyhedron is a Lipschitz polyhedron (X, d) such that for some triangulation T = (K, θ), there exist a measurable Riemannian metric g s = g ij dx i dx j on each maximal simplex s of i(|K|) (i as in Lemma 2.1), which satisfies almost everywhere in standard coordinate in the simplex s. Here the constant Λ is independent of a given simplex. The distance d g X on X is an intrinsic distance with respect to the metric g, meaning that d g = d g X is the infimal length of admissible path joining x to y. Actually (X, d g ) is a length space. The detailed definition is somewhat subtle and we refer to [EF01], for a careful discussion of Riemannian polyhedra.
A Riemannian metric g on a polyhedron X is said to be continuous, if relative to some (hence any) triangulation, g s is continuous up to the boundary on each maximal simplex s, and for any two maximal simplexes s and s ′ sharing a face t, g s and g s ′ induce the same Riemannian metric on t. There is a similar notion of a Lipschitz continuous Riemannian metric.
A Riemannian polyhedron has a well defined volume element given simplexwise by dµ g = det(g ij (x)) dx 1 dx 2 . . . dx n , this measure coincide with Hausdorff measure.

Further definitions.
A polyhedron X will be called admissible if in some (hence in any) triangulation, i) X is dimensionally homogeneous, i.e. all maximal simplexes have the same dimension n(= dim X), or equivalently every simplex is a face of some n-simplex and ii) X is locally (n − 1)-chainable, i.e. for every connected open set U ⊂ X, the open set U \X n−2 is connected. The boundary ∂X of a polyhedron X is the union of all non-maximal simplexes contained in only one maximal simplex. In this work we always assume that (X, g) satisfies ∂X = ∅.
By an n-circuit we mean a polyhedron X of homogeneous dimension n such that in some, (and hence any) triangulation, i) every (n − 1)-simplex is a face of at most two n-simplexes (exactly two if ∂X = ∅), and ii) X is (n − 1)-chainable, i.e. X\X n−2 is connected, or equivalently any two n-simplexes can be joined by a chain of contiguous (n − 1)-and n-simplexes.
Let S = S(X) denote the singular set of an n-circuit X, i.e. the complement of the set of all points of X having a neighborhood which is a topological n-manifold (possibly with boundary). S is a closed triangulable subspace of X of codimension ≥ 2, and X\S is a topological n-manifold which is dense in X. An admissible circuit is called a pseudomanifold. We call a pseudomanifold (X, g, d X ) a Lipschitz pseudomanifold, if g is Lipschitz continuous. If g is simplexwise smooth such that (X\S, g| X\S ) has the structure of a smooth Riemannian manifold, we call (X, g, d X ) a smooth pseudomanifold. 1 2.2. The Sobolev space W 1,2 (X). Let (X, g, d X ) denote an admissible Riemannian polyhedron of dimension n. We denote by Lip 1,2 (X) the linear space of all Lipschitz continuous functions u : (X, d X ) → R for which the Sobolev (1, 2)-norm u defined by u 2 1,2 = X (u 2 + |∇u| 2 ) dµ g = s∈S (n) (X) s (u 2 + |∇u| 2 ) dµ g is finite, S (n) (X) denoting the collection of all n-simplexes s of X, and |∇u| the Riemannian norm of the Riemannian gradient on each s. (The Riemannian gradient is defined a.e. in X or a.e. in each s ∈ S n (X), by Rademacher's theorem for Lipschitz functions on Euclidean domains.) The Lebesgue space L 2 (X) is likewise defined with respect to the volume measure. The Sobolev space W 1,2 (X) is defined as the completion of Lip 1,2 (X) with respect to the Sobolev norm · 1,2 . We use the notations Lip c (X), W 1,2 0 (X) and W 1,2 loc (X), for the linear space of functions in Lip(X) with compact support, the closure of Lip c (X) in W 1,2 (X) and all u ∈ L 2 loc (X) such that u ∈ W 1,2 (U ) for all relatively compact subdomains U in X.
Sobolev spaces on metric spaces. Here we recall a few basic notions on analysis on metric spaces. For the sake of completeness, we compare the L 2 based Sobolev space on admissible Riemannian polyhedra as in [EF01], with the one in [Che99], and show that they are equivalent. We use [Che99]  For any Lipschitz function f we have lip f (x) ≤ Lip f . In the special case Y = R n , if x is a point of differentiability of f , we observe that lip f (x) = Lip f (x) = |∇f (x)|. We now define the Soblolev space H 1,p , for 1 ≤ p < ∞.
where the infimum is taken over all sequence {g i }, for which there exists a sequence f i L p −→ f , such that g i is an upper gradient for f i , for all i.
For p ≥ 1, the Sobolev space, H 1,p , is the subspace of L p consisting of functions, f , for which f 1,p < ∞, equipped with the norm · 1,p . The space H 1,p is complete.
We define now the notions of generalized upper and minimal upper gradients. This will allow us to give a nice interpretation of the H 1,p norm of Sobolev functions. ii) For fixed p, a minimal generalized upper gradient for f is a generalized upper gradient g f , such that f 1,p = f L p + g f L p .
The following theorem ensures the existence of minimal generalized upper gradient for any Sobolev function.
Theorem 2.5. [Che99] For all 1 < p < ∞ and f ∈ H 1,p there exists a minimal generalized upper gradient, g f , which is unique up to modification on subsets of measure zero.
We will discuss two important properties of metric spaces called the ball doubling property and the Poincaré inequality for functions on them. These are essential assumptions to get a richer theory on metric spaces.
Definition 2.6. Let (Y, d, µ) be a metric measure space. The measure µ is said to be locally doubling if for all r ′ there exists κ = κ(r ′ ) such that for all y ∈ Y and 0 < r < r ′ 0 < µ(B r (y)) ≤ 2 κ µ(B r/2 (y)). (2) Definition 2.7. Let q ≥ 1. We say that Y supports a weak Poincaré inequality of type (q, p), if for all r ′ > 0, there exist constants 1 ≤ λ < ∞ and C = C(p, r ′ ) > 0 such that for all r ≤ r ′ , and all upper gradients g of f , where f x,r := Br(x) f dµ. If λ = 1, then we say that X supports a strong (q, p)-Poincaré inequality.
By Theorem 4.24 in [Che99], for any metric space which satisfies (11) and (12), for some 1 ≤ p < ∞ and q = 1, the subspace of locally Lipschitz functions is dense in H 1,p . Furthermore on a locally complete metric space with the mentioned properties, we have for some 1 < p < ∞ and for any f locally lipschitz, g f = Lip f , µ-almost everywhere (see [Che99] Theorem 6.1). Therefore, on a Riemannian polyhedron (X, g, µ g ), for any f ∈ H 1,2 , g f (y) = |∇f (y)| for a.e. y and it follows that H 1,2 is equivalent to W 1,2 . In the following, we always consider X = (X, g, µ g ) to be an admissible Riemannian polyhedron. Some of the concepts below are defined on metric spaces in general but for simplicity we present them only on Riemannian polyhedron and for p = 2. For more information on metric spaces we refer the reader to [BB11].
2.3. Potential theory background. In this subsection we recall some of the definitions in potential theory. First we define different notions of capacities (see [BB11]).
Sobolev and variational capacities. The Sobolev capacity of a set E ⊂ X is the number where the infimum is taken over all u ∈ W 1,2 (X) such that u ≥ 1 on E.
Let E ⊂ Ω ⊂ X and Ω bounded. The variational capacity is defined as where the infimum is taken over all u ∈ W 1,2 0 (Ω) such that u ≥ 1 on E. In this definitions the infimum can be taken only over u ≤ 1 such that it is equal 1 on a neighborhood of E. Also we write cap(E) = cap(E, X). Weakly harmonic and weakly sub/super harmonic functions. A function u ∈ W 1,2 loc (X) is said to be weakly harmonic if X ∇u, ∇ρ dµ g = 0 for every ρ ∈ Lip c (X).
A function u ∈ W 1,2 (X) is weakly harmonic if and only if u minimizes the energy E(v) among all functions v ∈ W 1,2 (X) such that v − u ∈ W 1,2 0 (X) (see [EF01]). In the following we discuss on the existence of minimizer under specific assumption on the Riemannian polyhedra.
Theorem 2.8. [EF01] Suppose the following Poincaré inequality holds: with c depending only on the admissible Riemannian polyhedron X. For any f ∈ W 1,2 (X) the class of competing maps contains a unique weakly harmonic function u. That function is the unique minimizer of As a corollary of the above theorem we have, Corollary 2.9. Assume that the domain Ω ⊂ X is bounded and such that the Sobolev capacity C(X\Ω) > 0. For any f ∈ W 1,2 (Ω), the class of functions has a unique solution u of the equation E(u) = E Ω , where , v − f ∈ W 1,2 0 (Ω)}. Let (X, g, d X ) denote an admissible Riemannian polyhedron of dimension n. We denote by Lip 1,2 (X) the linear space of all Lipschitz continuous functions u : (X, d X ) → R for which the Sobolev (1, 2)-norm u defined by is finite, S (n) (X) denoting the collection of all n-simplexes s of X, and |∇u| the Riemannian norm of the Riemannian gradient on each s. (The Riemannian gradient is defined a.e. in X or a.e. in each s ∈ S n (X), by Rademacher's theorem for Lipschitz functions on Euclidean domains.) The Lebesgue space L 2 (X) is likewise defined with respect to the volume measure. The Sobolev space W 1,2 (X) is defined as the completion of Lip 1,2 (X) with respect to the Sobolev norm · 1,2 . We use the notations Lip c (X), W 1,2 0 (X) and W 1,2 loc (X), for the linear space of functions in Lip(X) with compact support, the closure of Lip c (X) in W 1,2 (X) and all u ∈ L 2 loc (X) such that u ∈ W 1,2 (U ) for all relatively compact subdomains U in X.
Sobolev spaces on metric spaces. Here we recall a few basic notions on analysis on metric spaces. For the sake of completeness, we compare the L 2 based Sobolev space on admissible Riemannian polyhedra as in [EF01], with the one in [Che99], and show that they are equivalent. We use For any Lipschitz function We now define the Soblolev space H 1,p , for 1 ≤ p < ∞.
Definition 2.11. Whenever f ∈ L p (Y ), let where the infimum is taken over all sequence {g i }, for which there exists a sequence f i L p −→ f , such that g i is an upper gradient for f i , for all i.
For p ≥ 1, the Sobolev space, H 1,p , is the subspace of L p consisting of functions, f , for which f 1,p < ∞, equipped with the norm · 1,p . The space H 1,p is complete.
We define now the notions of generalized upper and minimal upper gradients. This will allow us to give a nice interpretation of the H 1,p norm of Sobolev functions.
Definition 2.12. i) The function, g ∈ L p is a generalized upper gradient for f ∈ L p , if there exist sequences, f i −→ g, such that g i is an upper gradient for f i , for all i. ii) For fixed p, a minimal generalized upper gradient for f is a generalized upper gradient g f , such that f 1,p = f L p + g f L p .
The following theorem ensures the existence of minimal generalized upper gradient for any Sobolev function.
Theorem 2.13. [Che99] For all 1 < p < ∞ and f ∈ H 1,p there exists a minimal generalized upper gradient, g f , which is unique up to modification on subsets of measure zero.
We will discuss two important properties of metric spaces called the ball doubling property and the Poincaré inequality for functions on them. These are essential assumptions to get a richer theory on metric spaces.
Definition 2.14. Let (Y, d, µ) be a metric measure space. The measure µ is said to be locally doubling if for all r ′ there exists κ = κ(r ′ ) such that for all y ∈ Y and 0 < r < r ′ 0 < µ(B r (y)) ≤ 2 κ µ(B r/2 (y)). (7) Definition 2.15. Let q ≥ 1. We say that Y supports a weak Poincaré inequality of type (q, p), if for all r ′ > 0, there exist constants 1 ≤ λ < ∞ and C = C(p, r ′ ) > 0 such that for all r ≤ r ′ , and all upper gradients g of f , where f x,r := Br(x) f dµ. If λ = 1, then we say that X supports a strong (q, p)-Poincaré inequality.
By Theorem 4.24 in [Che99], for any metric space which satisfies (11) and (12), for some 1 ≤ p < ∞ and q = 1, the subspace of locally Lipschitz functions is dense in H 1,p . Furthermore on a locally complete metric space with the mentioned properties, we have for some 1 < p < ∞ and for any f locally lipschitz, g f = Lip f , µ-almost everywhere (see [Che99] Theorem 6.1). Therefore, on a Riemannian polyhedron (X, g, µ g ), for any f ∈ H 1,2 , g f (y) = |∇f (y)| for a.e. y and it follows that H 1,2 is equivalent to W 1,2 . In the following, we always consider X = (X, g, µ g ) to be an admissible Riemannian polyhedron. Some of the concepts below are defined on metric spaces in general but for simplicity we present them only on Riemannian polyhedron and for p = 2. For more information on metric spaces we refer the reader to [BB11].
Proof. Since X satisfies the Poincaré inequality and using Theorem 5.54 in [BB11], Ω satisfies the inequality (5). By the above theorem, there is a unique minimizer which is weakly harmonic.
After correction on a null set every weakly harmonic function on X is Hölder continuous. A continuous weakly harmonic function is called harmonic.
Remark 1. From the discussion above one can see in the definition of variational capacity that there is a harmonic function u which takes the minimum in (4). This function is not necessarily continuous on the boundary of Ω\E.
Polar sets. A set S ⊂ X is said to be a polar set for the capacity if for every pair of relatively compact open sets U 1 ⋐ U 2 ⊂ X such that d(U 1 , X\U 2 ) > 0 we have According to Theorem 9.52 in [BB11] (see also section 3 in [GT02]), S is a polar set if and only if every point of X has an open neighborhood U on which there is a superharmonic function which equals +∞ at every point of S ∩ U . An equivalent formulation is to say that C(S) = 0 Lemma 2.16. A closed set S ⊂ X is a polar set if and only if for every neighborhood U of S and every ǫ > 0, there exists a function Lip(X) such that i) the support of ϕ is contained in X\S.
Proof. The proof is based on the definition of polar set and it is completely the same as the case of Riemannian manifolds. See Proposition 3.1 in [Tro99] for the proof of the equivalence on Riemannian manifolds.
The Dirichlet space L 1,2 0 (X). In this part we introduce the Dirichlet space L 1,2 0 (X) on an admissible Riemannian polyhedron X (see [EF01] for the definition of Dirichlet spaces). The Dirichlet space L 1,2 0 (X) determines a Brelot harmonic structure on X. Using this fact we can show, X has a symmetric Green function which gives us information on the singularities of X.
Proposition 2.17. [EF01] Suppose that, for every compact set K ⊂ X, for all x ∈ Lip c (X), with c(K) depending only on X and K. In particular, X is non-compact. The completion L 1,2 0 (X) of space Lip c (X) within L 1 loc (X) with respect to the norm E(u) 1/2 is then a regular Dirichlet space of strongly local type. L 1,2 0 (X) is a subset of W 1,2 loc (X).
For local questions, condition (9) is not required (it is automatically satisfied with X replaced by the open star of a point a of X relative to a sufficiently fine triangulation and in view of inequality (12)). As a consequence of Theorem 2.18, we have Proposition 2.19. The (n−2)-skeleton X (n−2) of an admissible Riemannian n-polyhedron is a polar set.
We should note that being polar is independent of the Riemannian structure on the polyhedron.

2.4.
Harmonic maps on Riemannian polyhedra. The energy of a map from a Riemannian domain to an arbitrary metric space was defined and investigated by Korevaar and Schoen [KS93]. Here, we give an introduction to the concept of energy of maps, energy minimizing maps and harmonic maps on a Riemannian polyherdron. In the case that the target Y is a Riemannian C 1 -manifold the energy of the map is given by the usuall expression (similarly when the target is a Riemannian polyhedron with continuous Riemannian metric).
Let (X, g) be an admissible n-dimensional Riemannian polyhedron with simplexwise smooth Riemannian metric. We do not require that g is continuous across lower dimensional simplexes. Let Y be an arbitrary metric space. Denote by L 2 loc (X, Y ) the space of all µ gmeasurable maps ϕ : X → Y having separable essential range 2 , and for which d Y (ϕ(·), q) ∈ L 2 loc (X, µ g ) for some point q (and therefore by the triangle inequality for any q ∈ Y ). For ϕ, ψ ∈ L 2 loc (X, Y ) define their distance The approximate energy density of a map ϕ ∈ L 2 loc (X, Y ) is defined for ε > 0 by The function e ε (ϕ) is of class L 1 loc (X, µ g ) (see [KS93]). Definition 2.20. The energy E(ϕ) of a map ϕ of class L 2 loc (X, Y ) is defined as We say that ϕ is locally of finite energy, and write ϕ ∈ W 1,2 loc (X, Y ), if E(ϕ| U ) < ∞ for every relatively compact domain U ⊂ X. For example every Lipschitz continuous map ϕ : X → Y is in W 1,2 loc (X, Y ). Now we give a necessary and sufficient condition for a map ϕ to be in W 1,2 loc (X, Y ).
Lemma 2.21. Let (X, g) be an admissible n-dimensional Riemannian polyhedron with simplexwise smooth Riemannian metric, and (Y, d Y ) a metric space. A map ϕ ∈ L 2 loc (X, Y ) is locally of finite energy if and only if there is a function e(ϕ) ∈ L 1 loc (X) such that e ε (ϕ) → e(ϕ) as ε → 0, in the sense of weak convergence of measures: Energy of maps into Riemannian manifolds. Let the domain be an arbitrary admissible Riemannian polyhedron (X, g) (g is only measurable with local elliptic bounds, unless otherwise specified), and the target is a Riemannian C 1 -manifold (N, h) without boundary, X of dimension n and Y of dimension m.
A chart η of N , η : V → R m is bi-Lipschitz if the components h αβ of h| V have elliptic bounds: Relative to a given countable atlas on a Riemannian C 1 -manifold (N, h), a map ϕ : (X, g) → (N, h) is of class W 1,2 loc (X, N ), or locally of finite energy, if i) ϕ is a quasicontinuous (after correction on a set of measure zero), ii) its components ϕ 1 , . . . , ϕ m in charts η : V → R m are of class W 1,2 (U ) for every quasiopen set U ⊂ ϕ −1 (V ) of compact closure in X, and iii) the energy density e(ϕ) of ϕ, defined a.e. in each of the quasiopen sets ϕ −1 (V ) covering X by e(ϕ) = (h αβ • ϕ)g(∇ϕ α , ∇ϕ β ), is locally integrable over (X, µ g ).
The energy of ϕ ∈ W 1,2 loc (X, N ) is defined by E(ϕ) = X e(ϕ) dµ g . There is also corresponding definition for the energy of maps into Riemannian polyhedra. There, (X, g) is admissible, dim X = n, and g is measurable with elliptic bounds on each n-simplex of X. The polyhedron Y is not required to be admissible, but its Riemannian metric h is assumed to be continuous.
Energy minimizing maps. We supLet (X, g, d X ) denote an admissible Riemannian polyhedron of dimension n. We denote by Lip 1,2 (X) the linear space of all Lipschitz continuous functions u : (X, d X ) → R for which the Sobolev (1, 2)-norm u defined by is finite, S (n) (X) denoting the collection of all n-simplexes s of X, and |∇u| the Riemannian norm of the Riemannian gradient on each s. (The Riemannian gradient is defined a.e. in X or a.e. in each s ∈ S n (X), by Rademacher's theorem for Lipschitz functions on Euclidean domains.) The Lebesgue space L 2 (X) is likewise defined with respect to the volume measure. The Sobolev space W 1,2 (X) is defined as the completion of Lip 1,2 (X) with respect to the Sobolev norm · 1,2 . We use the notations Lip c (X), W 1,2 0 (X) and W 1,2 loc (X), for the linear space of functions in Lip(X) with compact support, the closure of Lip c (X) in W 1,2 (X) and all u ∈ L 2 loc (X) such that u ∈ W 1,2 (U ) for all relatively compact subdomains U in X.
Sobolev spaces on metric spaces. Here we recall a few basic notions on analysis on metric spaces. For the sake of completeness, we compare the L 2 based Sobolev space on admissible Riemannian polyhedra as in [EF01], with the one in [Che99], and show that they are equivalent. We use [Che99]  For any Lipschitz function f we have lip f (x) ≤ Lip f . In the special case Y = R n , if x is a point of differentiability of f , we observe that lip f (x) = Lip f (x) = |∇f (x)|. We now define the Soblolev space H 1,p , for 1 ≤ p < ∞.
Definition 2.23. Whenever f ∈ L p (Y ), let where the infimum is taken over all sequence {g i }, for which there exists a sequence f i L p −→ f , such that g i is an upper gradient for f i , for all i.
For p ≥ 1, the Sobolev space, H 1,p , is the subspace of L p consisting of functions, f , for which f 1,p < ∞, equipped with the norm · 1,p . The space H 1,p is complete.
We define now the notions of generalized upper and minimal upper gradients. This will allow us to give a nice interpretation of the H 1,p norm of Sobolev functions.
Definition 2.24. i) The function, g ∈ L p is a generalized upper gradient for f ∈ L p , if there exist sequences, f i −→ g, such that g i is an upper gradient for f i , for all i. ii) For fixed p, a minimal generalized upper gradient for f is a generalized upper gradient g f , such that f 1,p = f L p + g f L p .
The following theorem ensures the existence of minimal generalized upper gradient for any Sobolev function.
Theorem 2.25. [Che99] For all 1 < p < ∞ and f ∈ H 1,p there exists a minimal generalized upper gradient, g f , which is unique up to modification on subsets of measure zero.
We will discuss two important properties of metric spaces called the ball doubling property and the Poincaré inequality for functions on them. These are essential assumptions to get a richer theory on metric spaces.
Definition 2.26. Let (Y, d, µ) be a metric measure space. The measure µ is said to be locally doubling if for all r ′ there exists κ = κ(r ′ ) such that for all y ∈ Y and 0 < r < r ′ 0 < µ(B r (y)) ≤ 2 κ µ(B r/2 (y)). (11) Definition 2.27. Let q ≥ 1. We say that Y supports a weak Poincaré inequality of type (q, p), if for all r ′ > 0, there exist constants 1 ≤ λ < ∞ and C = C(p, r ′ ) > 0 such that for all r ≤ r ′ , and all upper gradients g of f , where f x,r := Br(x) f dµ. If λ = 1, then we say that X supports a strong (q, p)-Poincaré inequality.
By Theorem 4.24 in [Che99], for any metric space which satisfies (11) and (12), for some 1 ≤ p < ∞ and q = 1, the subspace of locally Lipschitz functions is dense in H 1,p . Furthermore on a locally complete metric space with the mentioned properties, we have for some 1 < p < ∞ and for any f locally lipschitz, g f = Lip f , µ-almost everywhere (see [Che99] Theorem 6.1). Therefore, on a Riemannian polyhedron (X, g, µ g ), for any f ∈ H 1,2 , g f (y) = |∇f (y)| for a.e. y and it follows that H 1,2 is equivalent to W 1,2 . In the following, we always consider X = (X, g, µ g ) to be an admissible Riemannian polyhedron. Some of the concepts below are defined on metric spaces in general but for simplicity we present them only on Riemannian polyhedron and for p = 2. For more information on metric spaces we refer the reader to [BB11]. pose that (X, g), n-dimensional admissible Riemannian polyhedra with g simplexwise smooth and Y any metric space. A map ϕ ∈ W 1,2 loc (X, Y ) is said to be locally energy minimizing if X can be covered by relatively compact domains U ⊂ X for which E(φ| U ) ≤ E(ψ| U ) for every map ψ ∈ W 1,2 loc (X, Y ) such that ϕ = ψ a.e. in X\U .
Harmonic maps. Consider an admissible Riemannian polyhedron (X, g), of dimension n, and a metric space (Y, d Y ), Definition 2.28. A harmonic map ϕ : X → Y is a continuous map of class ϕ ∈ W 1,2 loc (X, Y ), which is locally energy minimizing in the sense that X can be covered by relatively compact subdomains U , for each of which there is an open set for every continuous map ψ ∈ W 1,2 loc (X, Y ) with ψ(U ) ⊂ V and ϕ = ψ in X\U . Every continuous, locally energy minimizing map ϕ : X → Y is harmonic. Also if Y is a simply connected complete Riemannian polyhedron of non-positive curvature, then a harmonic map ϕ : X → Y is the same as a continuous locally energy minimizing map.
We proceed now to characterize harmonic maps are continuous, weakly harmonic maps. We consider here (X, g) to be an arbitrary admissible Riemannian polyhedron and g just bounded measurable with local elliptic bounds, X of dimension n, and (N, h) a smooth Riemannian manifold without boundary, and the dimension of N is m. We denote by Γ k αβ the Christoffel symbols on N .
Definition 2.29. A weakly harmonic map ϕ : X → N is a quasicontinuous map of class W 1,2 loc (X, N ) with the following property: for any chart η : V → R n on N and any quasiopen set U ⊂ ϕ −1 (V ) of compact closure in X, the equation holds for every k = 1, . . . , m and every bounded function λ ∈ W 1,2 0 (U ). According to [EF01], a continuous map ϕ ∈ W 1,2 loc (X, N ) is harmonic (Definition 2.28) if and only if it is weakly harmonic (Definition 2.29).

Ricci Curvature on Riemannian Polyhedra
In the past few years, several notions of boundedness of Ricci curvature from below on general metric spaces have appeared. Sturm [Stu06] and Lott-Villani [LV09] independently introduced the so called curvature-dimension condition on a metric measure space denoted by CD (K, N ). The curvature dimension condition implies the generalized Brunn-Minkowski inequality (hence the Bishop-Gromov comparison and Bonnet-Myer's theorem) and a Poincaré inequality (see [Stu06,LV07,LV09]). Meanwhile, Sturm and Ohta introduced a measure contraction property denoted as M CP (K, N ) in Ohta's work. The condition M CP (K, N ) also implies the Bishop-Gromov comparison, Bonnet-Myer's theorem and a Poincaré inequality (see [Stu06,Oht07]). Note that all of these generalized notions of Ricci curvature bounded below are equivalent to the classical one on smooth Riemannian manifolds. Then after the reduced curvature dimension condition CD * (K, N ) has been introduce by Bacher and Sturm in [BS10] to overcome local-to-global property and it is equivalent to local CD(K, N ) condition. More recently the notion of Riemannian curvature dimension condition RCD(K, ∞) has been introduced in [AGS11b] and it is equivalent to the CD(K, ∞) on infinitesimally Hilbertian metric measure spaces. The finite dimensional version of this notion RCD(K, N ) has been introduced in [EKS13] and independently in [AMS13] and it is equivalent to CD * (K, N ) on infinitesimally Hilbertian metric measure spaces. Here we define both CD(K, N ), M CP (K, N ) and show that on a Riemannian polyhedron we can use both of them. In the following definitions, we always assume that (X, d) is a separable length space , P (X) is the set of all Borel probability measures µ satisfying X d X (x, y) 2 dµ(y) < ∞ for some x ∈ X. P 2 (X) is the set P (X) equipped with the L 2 -Wasserstein distance W 2 defined as For µ 0 , µ 1 in P 2 (X) and π in P (X × X) ranges between all transference plan between µ 0 and µ 1 . A transference plan is defined as as p 0 * (π) = µ 0 , p 1 * (π) = µ 1 p 0 , p 1 : X × X → X are projection to the first and second factores, respectively.

Curvature Dimension Condition:
We now define the notion of spaces satisfying CD(K, N ) condition following [LV09]. Suppose (X, d) is a compact length space. Let U : [0, ∞) → R be a continuous convex function with U (0) = 0. We define the non-negative function p(r) = rU ′ + (r) − U (r) with p(0) = 0. Given a reference probability measure ν ∈ P 2 (X), define the function is the Lebesgue decomposition of µ with respect to ν into an absolutely continuous part ρν and a singular part µ s , and If N ∈ [1, ∞) then we define DC N to be the set of such functions U so that is convex on (0, ∞). We further define DC ∞ to be the set of such functions U so that the function is convex on (−∞, ∞). A relevant example of an element in DC N is given by Definition 3.1. i) Given N ∈ [1, ∞], we say that a compact measured length space (X, d, ν) has non-negative N -Ricci curvature if for all µ 0 , µ 1 ∈ P 2 (X) with supp(µ 0 ) ⊂ supp(ν) and supp(µ 1 ) ⊂ supp(ν), there is some Wasserstein geodesic {µ t } t∈[0,1] from µ 0 to µ 1 so that for all U ∈ DC N and all t ∈ [0, 1], ii) Given K ∈ R, we say that (X, d, ν) has ∞-Ricci curvature bounded below by K if for all µ 0 , µ 1 ∈ P 2 (X) with supp(µ 0 ) ⊂ supp(ν) and supp(µ 1 ) ⊂ supp(ν), there is some Wasserstein geodesic {µ t } t∈[0,1] from µ 0 to µ 1 so that for all U ∈ DC ∞ and all t ∈ [0, 1], where λ : Note that inequalities (15) and (16) are only assumed to hold along some Wasserstein geodesic from µ 0 to µ 1 , and not necessarily along all such geodesics. This is what is called weak displacement convexity.
Proposition 3.2. If a compact measured length space (X, d, ν) has non-negative N -Ricci curvature for some N ∈ [1, ∞), then for all x ∈ supp(ν) and all 0 < r 1 ≤ r 2 the following inequality holds.
To generalize the notion of N -Ricci curvature to the non-compact case, we always consider a complete pointed locally compact metric measure space (X, ⋆, ν). Also for U ν to be a well-defined functional on P 2 (X), we impose the restriction ν ∈ M −2(N −1) , where M −2(N −1) is the space of all non-negative Radon measures ν on X such that We define M −∞ , by the condition X e −cd(x,⋆) 2 dν(x) < ∞, where c is a fixed positive constant. We should mention that most of the results for compact case (for example the Bishop-Gromov comparison) are valid for the non-compact case.

Measure Contraction Property
We define now the notion of measure contraction property M CP (K, N ) following [Oht07]. Let (X, d X ) be a length space, and µ a Borel measure on X such that 0 < µ(B(x, r)) < ∞ for every x ∈ X and r > 0, where B(x, r) denotes the open ball with center x ∈ X and radius r > 0.
Let Γ be the set of minimal geodesics, γ : [0, 1] → X, and define the evaluation map e t by e t (γ) := γ(t) for each t ∈ [0, 1]. We regard Γ as a subset of the set of Lipschitz maps Lip([0, 1], X) with the uniform topology. A dynamical transference plan Π is a Borel probability measure on Γ, and a path {µ t } t∈[0,1] ⊂ P 2 (X) given by µ t = (e t ) * Π is called a displacement interpolation associated to Π. For the exact definition of dynamical transference plan and displacement interpolation we refer the reader to [?]. For K ∈ R, we define the function Definition 3.3. For K, N ∈ R with N > 1, or with K ≤ 0 and N = 1, a metric measure space (X, d, µ) is said to satisfy the (K, N )-measure contraction property, the M CP (K, N ), if for every point x ∈ X and measurable set holds as measures on X, where we set 0/0 = 1 and, by convention, we read Here we state two results that we are going to use in the sequel.
Proposition 3.4. Let (M, g) be an n-dimensional, complete Riemannian manifold without boundary with n ≥ 2. Then a metric measure space (M, d g , ν g ) satisfies the M CP (K, n) if and only if Ric g ≥ K holds. Here d g and ν g denote the Riemannian distance and Riemannian volume element.
In the following theorem we state Bishop volume comparison theorem for the space satisfying M CP (K, N ).
Proposition 3.5. Let (X, µ) be a metric space satisfying the M CP (K, N ). Then, for any x ∈ X, the function . In the following we show that we can apply both measure contraction property and curvature dimension condition to a complete Riemannian polyhedra (X, g, µ g ). By previous section, a Riemannian polyhedron (X, g, µ g ) with the metric d X = d g X is a length space. Also for any x, y ∈ X we have e(x, y) ≤ d e X (x, y). where e(x, y) denotes the Euclidian distance. It is easy then to show that µ g is in M −2(N −1) and so on a complete Riemannian polyhedron we can use the notion of CD(K, N ). Also µ g is Borel and by Lemma 4.4 in [EF01], for any r there exist a constant c(r) such that c(r) −1 Λ −2n r n ≤ µ g (B(x, r)) ≤ c(r)Λ 2n r n for all x ∈ X. Therefore 0 < µ g (B(x, r)) < ∞ and the notion of M CP (K, N ) is also applicable here, for N ≥ n. (By Theorem 2.4.3 in [AT04], we have the Hausdorff dimension is n and by Corollary 2.7 in [Oht07] N should be greater than n.) In the rest of this work by Ric N,µg ≥ K can be taken to mean that (X, g, µ g ) satisfies either M CP (K, N ) or CD(K, N ) unless otherwise specified. In the following Lemma we show that any complete Riemannian polyhedron with non-negative Ricci curvature has infinite volume.
Lemma 3.6. Let (X, g, µ g ) be a complete, non-compact Riemannian polyhedron of dimension n. If Ric N,µg (X) ≥ 0, for N ≥ n, then X has infinite volume.
Proof. First we consider the case M CP (0, N ). By the Bishop comparison theorem, Theorem 3.5, for x ∈ X and all 0 < r 1 ≤ r 2 , By Proposition 10.1.1 in [Pap05], for every point in X, there exist a geodesic ray from that point. Consider a geodesic ray γ(t), 0 ≤ t < ∞, such that γ(0) = x. We construct the balls B(γ(t), t − 1) and B(γ(t), t + 1) centered at γ(t) with radius t − 1 and t + 1. We have Letting t → ∞, we get µ g (B(γ(t), t − 1)) → ∞ and therefore X has infinite volume. By Theorem 3.2, and since X is a complete locally compact length space, we can repeat the proof for the case when X satisfies the non-negative N -Ricci curvature condition CD(0, N ), for N ∈ (1, ∞).
Here we recall some remarks that we need through the rest of this paper.
Remark 3. 1.By Remark 5.8 in [Stu06] if (X, d, µ) satisfy M CP (K, N ) so does any convex set A ⊂ X. When X is a smooth pseudomanifold, for any point x ∈ X\S, there exist a closed totally convex neighborhood V around x (for every point in a Riemannian manifold there is a geodesic ball which is totally convex). Therefore if X satisfies Ric N,µg ≥ K, so does X\S. The same result is valid on metric measure spaces with CD(K, N ) condition (see Theorem 5.53 in [LV09]).

Some Function Theoretic Properties On Complete Riemannain Polyhedra
4.1. Liouville-type Theorems for Functions. The aim of this section is to generalize some of the results in [Yau82] in order to prove some vanishing theorems for harmonic maps on Riemannian polyhedra. In [Yau82], Yau used the Gaffney's Stokes theorem on complete Riemannian manifolds to prove that every smooth subharmonic function with bounded ∇f L 1 is harmonic. He uses this fact to prove that there is no non-constant L p , p > 1, non-negative subharmonic function on a complete manifold. We generalize this theorem to complete admissible polyhedra for p = 2 which is the content Theorem 1.1 Proof of Theorem 1.1. Fix a base point x 0 ∈ X and define ρ : X → R as Observe that ρ is 1 R -Lipschitz and ρ = 0 on X\B(x 0 , 2R) and ρ = 1 on B(x 0 , R). Since f is subharmonic, Combining the two previous inequality, we obtain The last line is a polynomial, P (ψ) = ψ 2 − 2bψ + c, where ψ is and it has non-positive value only if b 2 ≥ c, which means that Sending R to infinity and using the fact that f has finite L 2 -norm, we conclude that X |∇f | 2 dµ g = 0.
Since X is admissible, f is constant on X. (First we prove that f is constant on each maximal n-simplex S and then using the n − 1-chainability of X, we prove this in the star of any vertex p of X and then by connectedness on X.) In the following theorem, we show that the Laplacian of a weakly subharmonic function f ∈ W 1,2 loc (X) on a pseudomanifold in the distributional sense is a locally finite Borel measure. This gives us a verifying of Green's formula on these spaces. We then use this theorem, to prove that a continuous weakly subharmonic function with ∇f L 1 < ∞ on a complete normal circuit is harmonic.
Theorem 4.1. Let (X, g, µ g ) be an n-pseudomanifold. Let f be a weakly subharmonic function in W 1,2 loc (X), such that ∇f L 1 is finite. Then there exists a unique locally finite Borel measure m f on X such that Proof. We consider the Lipschitz manifold M = X\S and the chart {(U α , ψ α )} on M . We show that is a linear continuous functional on D α = Lip c (ψ α (U α )) with respect to the topology of uniform convergence on compact sets. The linearity is obvious. We have and so Λ α is continuous. Since Lip c (U ) is dense in C c (U ) for a locally compact domain U , see Proposition 1.11 in [BB11], then Λ α is also continuous on C c (ψ α (U α )). By assumption f is subharmonic and so Λ α is positive. By Riesz representation theorem, Λ α is a unique positive Radon measure. It follows that there is a positive Radon measure m α such that Now we consider the partition of unity {ρ α } subordinate to {U α }. We put m = α ρ α ψ * (m α ) and we define m f (U ) = m(U \S) on each Borel set U . Obviously m f is positive and locally finite. The uniqueness comes from the uniqueness of m α .
We recall a remark concerning the above theorem.
Remark 4. Gigli introduced the notion of Laplacian as a set of locally finite Borel measure (see Definition 4.4 in [Gig12]). There he proved that on infinitesimally Hilbertian spaces this set contains only one element.
In the smooth setting, as a corollary of Gaffney's Stokes theorem, we have that on a complete Riemannian manifold every smooth subharmonic function f with bounded ∇f L 1 is harmonic. We generalized this theorem on pseudomanifolds in Theorem 1.2.
Proof of Theorem 1.2. We put A 1 = ∇f L 1 . We consider a sequence of cut-off functions ρ n for fixed q ∈ X such that ρ n is 1 n -Lipschitz and such that ρ n is equal to 1 on B(q, R) and its support is in B(q, R + n). f is a subharmonic function which satisfies the condition of previous lemma, so there is a unique Borel measure m f such that Let h be any function in Lip c (X) with support in B(q, R). We have and tending n to infinity, we have and implying that f is harmonic.
Now we prove a generalization of Proposition 2 in [Yau82], see Theorem 1.3 for the exact statement. We give here another proof of the theorem above for smooth pseudomanifolds under the extra assumption that f should have finite energy. Instead of Theorem 4.1, we goal Cheeger's Green formula on compact smooth pseudomanifolds in the proof.
Proof of Theorem 1.3. We put A 1 = ∇f L 1 and A 2 = ∇f L 2 . We present the proof in several steps.
Step 1. We consider a sequence of cut-off functions ρ n as above such that the support of ρ n is in B(q, R + n) for fixed q ∈ X\X n−2 and some R and ρ n is equal to 1 on B(q, R) and ρ n is 1 n -Lipschitz.
Step 2. The (n−2)-skeleton in X, X n−2 , is a polar set. We consider, shrinking bounded neighborhoods U j of X n−2 in B(q, R + j), such that in each B(q, R + j), we have By the definition of polar set, for the open domains U j and U j−1 , we have cap(X n−2 ∩ U j , U j−1 ) = 0. This means that for every j, there exists a function ϕ j ∈ Lip(X) such that ϕ j ≡ 1 in a neighborhood of X n−2 ∩ U j and ϕ j is zero outside U j−1 and X |∇ϕ j | 2 < 1 j . Moreover we have 0 ≤ ϕ j ≤ 1.
We put η j = 1 − ϕ j . The function η j has the property that the closure of its support, supp η j , is contained in X\X n−2 and the set K j = supp η j ∩ B(q, R + j) is compact. Furthermore K j s make an exhaustion of M = X\X n−2 .
Step 3. According to Theorem 2 in [GW79], for any j, there exist a smooth subharmonic function f j on M such that sup x∈K j |f j (x) − f (x)| < 1 j and |∇f j (x)| ≤ |∇f (x)| on K j .
Step 4. In this step we prove M ∆f j · ξ j dµ g = − M ∇f j , ∇ξ j dµ g .
where ξ j = ρ j · η j . To prove the above equality, first we recall a Remark from [Che80].
Let h be any smooth function with compact support in M ∩ B(q, R). Then there is a K m such that the support of h is in B(q, R) ∩ K m . For j large enough we will have ξ j ≡ 1 on K m and so we have 0 ≤ B(q,R)∩Km ∆f j dµ g ≤ 1 j (A 2 + A 1 ).
considering the formula (19) as above, for j large enough we have Letting j go to infinity, we got M ∆h · f dµ g = 0. By use of Weyl's lemma f is a smooth harmonic function on M .
Step 6. In this step we prove f is locally Lipschitz. Since f is in f ∈ W 1,2 loc (X) and by Theorem 12.2 in [BB11], f is harmonic on X. Then by Corollary 6 in [Kel13] (see also Theorem 3.1 in [Jia13]) f is locally Lipschitz.
Remark 6. In [KRS03], the Lipschitz regularity of harmonic functions has been proved on metric measure spaces under the assumptions of Ahlfors regularity of the measure, Poincaré inequality and a heat semigroup type curvature condition. In the most recent work of [Kel13,Jia13,Jia12] the Lipschitz regularity of the functions whose Laplacian are either in L p or in L ∞ has been studied under more relaxed assumption on the measure. Furthermore the Cheng-Yau gradient estimate has been obtained in [HKX13] for metric measure spaces under RCD(K, N ) curvature dimension condition. See [GKO13,SZ11] for the equivalent results on Alexandrov spaces.
Step 7. Now we show f is constant. Since M has non-negative Ricci curvature, by the Bochner formula |∇f | is subharmonic on M and so on X (see Theorem 12.2 in [BB11]). By Lemma 1.1, |∇f | is constant. Since the L 2 -norm of |∇f | is finite we have |∇f | ≡ 0. By Lemma 3.6, f should be constant.

Vanishing Results for Harmonic Maps on Complete Smooth
PseudoManifolds. In this subsection we prove Theorems 1.4 and 1.5.
Proof of Theorem 1.4. By Remark 3, we know that on the Riemannian manifold M = X\S we have non-negative Ricci curvature. We show that for ǫ > 0, e(u) + ǫ is weakly subharmonic on X. As the restriction maps u = u| M : (M, g) → Y is harmonic, we have a Bochner type formula for harmonic map on M and ∆e(u) > |B(u)| 2 , where B(u) is the second fundamental form of the map u. Also by Cauchy-Schwarz we have, |∇e(u)| 2 ≤ 2e(u)|B(u)| 2 , and so for ǫ > 0, on X\S ∆ e(u) + ǫ ≥ 0. See e.g. the calculation in [Xin96] Theorem 1.3.8. Thus e(u) + ǫ is subharmonic on X\S and by Theorem 12.2 in [BB11], subharmonicity on X follows since S is polar and e(f ) is locally bounded. Therefore, X ∇ e(u) + ǫ, ∇ρ dµ g ≤ 0 ρ ∈ Lip c (X).
As in the proof of Theorem 1.1, Note that e(u) + ǫ satisfies all the assumptions of the Theorem 1.1, except the finiteness of L 2 -norm which we do not need in this step. Set B ′ R = B R \{x ∈ B R , e(u)(x) = 0}. Then Letting ǫ → 0 gives and letting R → ∞ and by finiteness of the energy we have which implies that e(u) is constant. If e(u) is not zero everywhere this means that the volume of X is finite. By Lemma 3.6, this is impossible and so u is constant. Now we prove Theorem 1.5. By the following lemma, the function d(u(·), q), where q is an arbitrary point in Y , is subharmonic under suitable assumption on the curvature of Y . We refer the reader to [EF01] Lemma 10.2, for the proof.
Lemma 4.2. Let (X, g) be an admissible Riemannian polyhedron, g simplexwise smooth. Let (Y, d Y ) be a simply connected complete geodesic space of non-positive curvature, and let u ∈ W 1,2 loc (X, Y ) be a locally energy minimizing map. Then u is a locally essentially bounded map and for any q ∈ Y , the function d(u(·), q) of class W 1,2 loc (X, Y ) is weakly subharmonic and in particular essentially locally bounded.

We have
Proof of Theorem 1.5. According the lemma above the function v(x) = d(u(x), u(x 0 )) for some x 0 ∈ X, is weakly subharmonic. We know that |∇v| 2 ≤ ce(u), where c is a constant. v is a continuous subharmonic function whose gradient is bounded by an L 1 and L 2 integrable function. According to Lemma 1.1, v is a constant function and so u is a constant map.
Remark 7. Using above argument we have also showed every continuous harmonic map u : (X, g) → Y belonging to W 1,2 loc (X, Y ) with M e(u) dµ g < ∞, where (X, g, µ g ) is a complete, non-compact n-pseudomanifold with non-negative n-Ricci curvature CD(0, n) and Y a simply connected, complete geodesic space of non-positive curvature is Lipschitz continuous.

2-Parabolic Riemannian Polyhedra
In this last section we prove Liouville-type theorems for harmonic maps defined on a Riemannian polyhedra X without any completeness or Ricci curvature bound assumption. We assume instead X to be 2-parabolic. Some of these results extend known results for the case of Riemannian manifolds. As for Riemannian manifolds, we say that a domain Ω ⊂ X in an admissible Riemannian polyhedra X is 2-parabolic, if cap(D, Ω) = 0 for every compact set D in Ω, otherwise 2-hyperbolic. A reference on this subject is [GT02], where the notion is discussed for general metric measure spaces. Our main results in this section are Theorems 1.7 and 1.8. We will first need the following characterization of 2-parabolicity.
Lemma 5.1. The domain Ω is 2-parabolic if and only if there exists a sequence of functions ρ j ∈ Lip c (Ω) such that 0 ≤ ρ j ≤ 1, ρ j converges to 1 uniformly on every compact subset of Ω and Ω |∇ρ j | 2 dµ g → 0.
Proof. First suppose Ω is 2-parabolic. Then every compact set D ⊂ Ω, with nonempty interior satisfies cap(D, Ω) = 0. We choose an exhaustion D ⊂ D 1 ⊂ D 2 ⊂ . . . ⊂ Ω of Ω by compact subsets such that cap(D j , Ω) = 0 for all j. Hence we can find the function ρ j ∈ Lip c (Ω) (using the fact that Lip c (Ω) is dense in W 1,2 0 (Ω)) such that ρ j ≡ 1 on D j and Ω |∇ρ j | 2 dµ g ≤ 1/j 2 . We have constructed the desired sequence ρ j . Conversely, suppose there exists, a sequence ρ j ∈ Lip c (Ω) with the stated properties. Then we can find a compact subset B ⊂ Ω and j 0 such that ρ j ≥ 1/2 for every j ≥ j 0 . It follows that cap(B, Ω) = 0 The following lemma shows that the 2-parabolicity remains after removing the singular set of a Riemannian polyhedron.
Lemma 5.2. If X is a 2-parabolic admissible Riemannian polyhedron and E ⊂ X is a polar set, then Ω := X\E is 2-parabolic.
Proof. X is 2-parabolic, so by Lemma 5.1, there are an exhaustion of X and a sequence of function ρ j ∈ Lip c (X) such that 0 ≤ ρ j ≤ 1 and ρ j → 1 uniformly on each compact set, and X |∇ρ j | 2 dµ g → 0. Also by Lemma 2.16, there exist another sequence of functions ϕ j with support in X\E such that ϕ j → 1 on each compact set of X\E and X |∇ϕ j | 2 dµ g → 0. The functions ρ j ϕ j on Ω provides the condition for 2-parabolicity in Lemma 5.1.
The following result is an extension of Theorem 5.2 in [Hol90] to admissible Riemannian polyhedra (see Proposition 1.9).
Proof of Proposition 1.9. Since f is continuous, for any ǫ and at any point x 0 in X there exist a relatively compact neighborhood B 0 of x 0 such that f (x) > f (x 0 ) − ǫ on B 0 . X is 2-parabolic, so cap(B 0 , X) = 0. Consider an exhaustion of X by regular domains U i such that B 0 ⋐ U 1 ⋐ U 2 ⋐ . . . ⋐ X. By Corollary 11.25 in [BB11], such exhaustion exists.
There exist functions u i which are harmonic on U i \B 0 , u i ≡ 1 on B 0 and u i ≡ 0 on X\U i (See [GT01] and also Lemma 11.17 and 11.19 in [BB11]). The maximum principle (see Theorem 5.3 in [EF01] or Lemma 10.2 in [BB11] for the comparison principle) implies that 0 ≤ u i ≤ 1 u i+1 ≥ u i on U i .

Define the function
On the other hand f ≥ h i on the boundary of U i \B 0 . By the comparison principle f ≥ h i in U i \B 0 , so f ≥ f (x 0 ) − ǫ on X. Letting ǫ → 0, we obtain f ≥ f (x 0 ) on X. If f is non-constant, there exist x 1 ∈ X with f (x 1 ) > f (x 0 ). By the same argument we obtain f > f (x 1 ). This is a contradiction and thus f is constant.
We prove the analogue of Theorem 1.3, for 2-parabolic admissible Riemannian polyhedra. This is the content of Proposition 1.11.
By Lemma 4.1, there is a locally finite Borel measure m f such that Now let h be an arbitrary test function in Lip c (X) where its support is in D. We have and so f is harmonic on X.
Similarly we have the following result generalizing Theorem 5.9 in [Hol90].
Let U j be an exhaustion of X by regular domains U j ⊂ U j+1 ⋐ X. There is a continuous function u i,j such that u i,j are harmonic on U j and u i,j = f i in X\U i . Also u i,j is continuous on X and ∇u i,j L 2 is finite. We have −i ≤ u i,j ≤ i . According to Theorem 6.2 in [EF01], u i,j are Hölder continuous (after correction on a null set), and since they are uniformly bounded, by Theorem 6.3 in [EF01], they are locally uniformly Hölder equicontinuous and by Theorem 9.37 in [BB11], there is a subsequence which converges locally uniformly to some u i as j → ∞. (Note that the definition of harmonicity as in [BB11] is consistent with our definition.) The function u i is bounded and harmonic and hence is constant. Moreover u i,j − f i ∈ L 1,2 0 and so f i ∈ L 1,2 0 . Therefore X |∇f | 2 dµ g = lim i→∞ X ∇f, ∇f i dµ g = 0, and f is constant.
By use of Lemma 4.2 and the above propositions, the proofs of Corollaries 1.7 and 1.8 are straightforward.