A formal Riemannian structure on conformal classes and the inverse Gauss curvature flow

We define a formal Riemannian metric on a given conformal class of metrics on a closed Riemann surface. We show interesting formal properties for this metric, in particular the curvature is nonpositive and the Liouville energy is geodesically convex. The geodesic equation for this metric corresponds to a degenerate elliptic fully nonlinear PDE, and we prove that any two points are connected by a $C^{1,1}$ geodesic. Using this we can define a length space structure on the given conformal class. We present a different approach to the uniformization theorem by studying the negative gradient flow of the normalized Liouville energy, a new geometric flow whose principal term is the inverse of the Gauss curvature. We prove long time existence of solutions with arbitrary initial data and weak convergence to constant scalar curvature metrics. This is all a special case of a more general construction on even dimensional manifolds related to the $\sigma_{\frac{n}{2}}$-Yamabe problem, which will appear in a forthcoming article.


Introduction
In this paper we define a formal Riemannian metric on the set of metrics in a conformal class with positive (or negative) curvature. Namely, let (M, g 0 ) be a compact Riemannian surface with positive Gauss curvature K 0 > 0, and let [g 0 ] denote the conformal class of g 0 . Define Γ + 1 = {g u = e 2u g 0 ∈ [g 0 ] ∶ K u = K gu > 0}, (1.1) the space conformal metrics with positive Gauss curvature. Formally, the tangent space to [g 0 ] at any metric g u ∈ [g 0 ] is given by C ∞ (M ). Let K u denote the Gauss curvature of g u ∈ Γ + 1 . We define for φ, ψ ∈ C ∞ (M ) (cf. Definition 2.3), ⟪φ, ψ⟫ u = M φψK u dA u , (1.2) where dA u is the area form of g u . In other words, we weight the standard L 2 metric with the Gauss curvature of the given conformal metric. If the Gauss curvature of g 0 is negative, we define Γ − 1 = {g u = e 2u g 0 ∈ [g 0 ] ∶ K u = K gu < 0}, (1.3) and the metric associated to this space is given by This definition is loosely inspired by the Mabuchi-Semmes-Donaldson [17,23,10] metric of Kähler geometry, wherein a formal Riemann metric is put on a Kähler class by imposing on the tangent space to a given Kähler potential the L 2 metric with respect to the associated Kähler metric. As observed in [17], this metric enjoys many nice formal properties, for instance nonpositive sectional curvature. Moreover, it has a profound relationship to natural functionals in Kähler geometry such as the Mabuchi K-energy and the Calabi energy, as well as their gradient flow, the Calabi flow. Based on these excellent formal properties Donaldson proposed a series of conjectures on the existence of geodesics, geodesic rays, as well as the existence properties of the Calabi flow. The tremendous work of many authors (an incomplete list of references is [3,4,5,6,7,9,13,20,22]) has resulted in the verification of many of these conjectures, which largely centers around a detailed analysis of the very delicate geodesic equation, which can be interpreted as a degenerate Monge-Ampere equation.
As we will see, there is a tight analogy in many respects between the Mabuchi metric and the metric defined in (1.2). In section 2 we establish a formal path derivative which can be regarded as the Levi-Civita connection associated to the metric. Using these we compute the sectional curvature, and show that the metric is nonpositively curved. Next, in section 3 we derive the geodesic equation. Formal calculations derived using either the path derivative or variations of the length functional yield that a one-parameter family of conformal factors u ∶ [a, b] → Γ + 1 is a geodesic if and only if We end section 3 with the fundamental observation that one parameter families of conformal transformations are automatically geodesics (Proposition 3.5).
Section 4 contains the proof of the existence of C 1,1 geodesics connecting any two points in Γ + 1 . Equation (1.5) turns out to be a fully nonlinear, degenerate elliptic equation. We study a natural regularization of (1.5) which renders it a convex, strictly elliptic equation. By maximum principle arguments we establish a priori C 1,1 estimates independent of the regularization parameter, which yield the existence of the C 1,1 solution as claimed. With this in place in section 5 we rigorously show that the length of the unique regularizable geodesic connecting any two points does indeed define a metric space structure (Γ + 1 , d) (Corollary 5.6), and that this metric space is nonpositively curved in the sense of Alexandrov (Proposition 5.8). The situation is summarized in the following theorem.
Theorem 1.1. Let (M 2 , g 0 ) be a compact Riemann surface. Then (Γ ± 1 , d) is a length space, with any two points connected by a unique regularizable C 1,1 geodesic. Moreover, it is nonpositively curved in the sense of Alexandrov.
Furthering the analogy with the Kähler setting, the metric (1.2) is closely associated with the gradient flow of the normalized Liouville energy. Previously Osgood-Phillips-Sarnack [19] studied the negative gradient flow, but with respect to the L 2 metric, yielding an equation which is similar to Ricci flow. With the ambient geometry given by the weighted L 2 metric on Γ + 1 , we arrive at a different evolution equation, expressed in terms of the conformal factor as where K is the average Gauss curvature. This is a fully nonlinear parabolic equation for u. On Γ − 1 we arrive at Generically we will refer to these as inverse Gauss curvature flow. Our primary results are as follows: Theorem 1.2. Fix (M 2 , g) a compact Riemann surface and u ∈ Γ ± 1 . (1) The solution to IGCF with initial condition u exists on [0, ∞).
(2) The normalized Liouville energy is convex in time along the flow line, i.e. (3) Given v(x, t) another solution to IGCF, the distance between flow lines is nonincreasing, i.e. d dt d(u(t), v(t)) ≤ 0.
(4) If u ∈ Γ − 1 , then the solution converges as t → ∞ in the C ∞ topology to the unique conformal metric of constant scalar curvature. (5) If u ∈ Γ + 1 and (M 2 , g) ≅ (S 2 , g S 2 ), then the solution converges weakly in the distance topology to a minimizer for F in the completion (Γ + 1 , d).
Remark 1.3. Properties (2) and (3) are directly analogous to results relating the K-energy, Mabuchi metric, and Calabi flow (cf. [4]). We emphasize that the point of the hypothesis (M 2 , g) ≅ (S 2 , g S 2 ) is that we are NOT yet able to use the IGCF to provide an a priori proof of the Uniformization Theorem. We require the existence of a constant scalar curvature metric to ensure the convergence of the flow in the distance topology. Remark 1.4. Although our results are in the setting of two dimensions, this is actually a special case of a more general construction on even dimensional manifolds. In dimensions n ≥ 4, one can define a Riemannian structure on subsets of conformal classes satisfying an admissibility condition which naturally arises in the study of the σ n 2 -Yamabe problem. As in the case of surfaces, the underlying metric is closely associated to a functional whose critical points 'uniformize' the conformal class. This will be presented in a forthcoming article [14].

Metric, connection, and curvature
In this section we define the formal Riemannian metric on the space of conformal metrics of positive/negative curvature. Because of the dependence on the sign of the curvature, we will first consider the positive case in detail, then provide the corresponding results for metrics of negative curvature without (or at most cursory) proofs.
2.1. The positive cone. Let (M, g 0 ) be a closed surface with positive Gauss curvature, and let denote the space conformal metrics with positive Gauss curvature. The formal tangent space at g u is where dA u is the volume form of the metric g u = e 2u g 0 . Remark 2.2. To simplify notation, we will often write u ∈ Γ + 1 to mean g u = e 2u g 0 ∈ Γ + 1 . Given a path of conformal factors u ∶ [a, b] → Γ + 1 and a vector field α = α(⋅, t) along u, we define where ⟨⋅, ⋅⟩ u denotes the inner product with respect to g u . Proof. Let α, β be vector fields along the path u ∶ [a, b] → Γ + 1 . To simplify notation we will drop the subscript u, and all metric-dependent quantities (curvature, area form, etc.) will be understood to be with respect to g u .
We first prove compatibility. This will require us to record the standard variational formulas for a path of conformal metrics g = g(t) = e 2u g 0 : (2.5) To compute the torsion, let u = u(⋅, s, t) be a two-parameter family of conformal factors in Γ + 1 . Then D ∂s Proposition 2.4. Given φ, ψ ∈ T u Γ + 1 , the sectional curvature of the plane in T u Γ + 1 spanned by φ, ψ is given by Proof. Let u = u(s, t) be a 2-parameter family of conformal factors, and α = α(s, t) ∈ T u(s,t) Γ + 1 . Using the formula for the connection, we have D ∂s (2.7) In the following, we will omit the subscript u, and all metric-dependent quantities will be understood to be with respect to g u . To evaluate I, we will need the variational formulas (2.5) along with (2.9) Turning to II, we write Combining (2.9) and (2.10) and skew-symmetrizing in s, t, we have D ∂s To compute the sectional curvature of the plane spanned by { ∂u ∂s , ∂u ∂t }, we take α = ∂u ∂t in the formula above, then take the inner product with ∂u ∂s : If we integrate by parts in the last two terms, we find as claimed.
2.2. The negative cone. Now assume (M, g 0 ) is a closed surface with K 0 < 0, and let denote the space conformal metrics with negative Gauss curvature. The formal tangent space at g w is where dA w is the area form of the metric g w = e 2w g 0 .
As before, we write w ∈ Γ − 1 to mean g w = e 2w g 0 ∈ Γ − 1 . Given a path of conformal factors w ∶ [a, b] → Γ + 1 and a vector field α = α(⋅, t) along u, we now define The proof of the next two results are essentially the same as in the case of the positive cone: Lemma 2.6. The connection defined by (2.18) is metric-compatible and torsion-free.

Geodesics, length, and energy
Using the definition of the positive cone metric in (2.3) we can also define the associated notions of energy and length.
The energy density is The length of u is By taking the first variation of the energy we arrive at the geodesic equation: 1 is a geodesic, and write g u = e 2u g 0 , where g 0 ∈ Γ + 1 . By the Gauss curvature equation, where K 0 is the Gauss curvature of g 0 . Therefore, we can rewrite (3.5) as As we will see in Section 4, this is a degenerate elliptic fully nonlinear PDE.
In the next lemma we show two basic properties of geodesics. As a preface, we remark that there is a canonical isometric splitting of T Γ + 1 with respect to the metric. In particular, the real line R ⊂ T u Γ + 1 given by constant functions is orthogonal to We will see that geodesics preserve this isometric splitting, and are automatically parameterized with constant speed: Proof. Differentiating, integrating by parts, and using the geodesic equation gives with p >> 1 large and apply (3.8), then in the limit as p → ∞ we have the following corollary of Lemma 3.3: 3.1. Example: The round sphere. Let (S 2 , g 0 ) denote the round sphere. Using stereographic projection σ ∶ S 2 ∖ {N } → R 2 , where N ∈ S 2 denotes the north pole, one can define a one-parameter of conformal maps of S 2 by conjugating the dilation map δ α ∶ x ↦ α −1 x on the plane with σ: Taking α(t) = e λt , where λ is a fixed real number, we can define the path of conformal metrics where ξ = x 3 is the coordinate function (see [16]).
Proof. By (3.10), Letting subscripts denote differentiation in t, we have and hence Since α = e λt , this simplifies to Also, if ∇ denotes the connection with respect to the round metric, Using the fact that ξ satisfies it follows from (3.12) and (3.13) that (3.14) Since K u = 1 for all t, comparing (3.11) and (3.14) we see that u satisfies the geodesic equation (3.5).

3.2.
Geodesics in the negative cone. Analogous to the definitions for the positive cone, the energy and length of a path w ∶ [a, b] → Γ − 1 are By taking the first variation of E we arrive at the geodesic equation for the negative cone: We also have the basic properties of Lemma 3.3:

Existence of geodesics
In this section we prove a priori estimates for solutions of the geodesic equation in the positive and negative cones. We begin by introducing a regularization of the geodesic equation, then prove estimates for derivatives up to order two which are independent of the regularizing parameter. Using a continuity argument, we show that classical solutions of the regularized equation exist and are unique. This allows us to define the notation of a regularizable geodesic (see Definition 4.19).
We begin with estimates for a geodesic u in the positive cone. By Lemma 3.3, we may assume u ∶ [0, 1] → Γ + 1 . Recall from (3.7) that u satisfies the PDE where u 0 , u 1 ∈ Γ + 1 . To simplify the notation, in the following we will denote derivatives with respect to t by subscripts, and we will omit the subscript 0: all metric-dependent quantities are with respect to the background metric g 0 .
As we will see, (4.1) is degenerate elliptic, so it will be necessary to regularize the equation. To simplify some of the estimates, and to clarify the dependence on the boundary data and other parameters, we will choose a fairly specific regularization.
Define the operator (4.5) then u ∈ Γ + 1 and solves (4.1). In Lemma 4.2 we will see that (4.4) is elliptic when f > 0, but degenerate elliptic when f = 0. Therefore, to prove the existence of solutions of the geodesic equation, we proceed as follows: (1) Given admissible boundary data u 0 , u 1 , we prove the existence of an admissible solutionũ to (4.4), for a specific choice of f = f 0 > 0.
(2) For 0 < ǫ ≤ 1 we establish a priori estimates for solutions of with f = ǫf 0 , and subject to the given boundary conditions.
(4) Taking the limit as ǫ → 0, we obtain a solution u of the geodesic equation (4.1).
For the first step, letũ where A > 0 will be specified later. We easily calculatẽ u tt = −2A, (4.7) Since u 0 and u 1 are admissible, there is a δ 0 > 0 such that (4.9) and it follows thatũ is admissible. By (4.7), Choosing A = A 0 > 0 large enough (depending only on the boundary data), we have Therefore,ũ satisfiesũ with f 0 > 0. We have thus proved Lemma 4.1. Given admissible boundary data u 0 , u 1 , there is a function f 0 > 0 and an admissible solution u =ũ of (4.4) with f = f 0 . The next step is to prove C 2 -estimates for admissible solutions u of (⋆ ǫ ), where f = ǫf 0 , and u satisfies the given (admissible) boundary data. To this end, let L denote the linearized operator, defined by (4.13) Proof. Fix a point (x 0 , t 0 ) and in normal coordinates at this point let α = ∇u t (x 0 , t 0 ) and κ = (−∆u + K)(x 0 , t 0 ). Then the symbol of L is given by Using the equation, we can substitute for u tt and write

Derived Equations.
Before proceeding to the estimates we begin with some preliminary calculations. We emphasize that in this subsection we will not make use of the fact that f = ǫf 0 ; we will only need that f > 0.
Let u be an admissible solution of (⋆ ǫ ), and define Then Proof. This is a straightforward calculation.
Proof. We compute and substituting this into (4.15) gives the result.
Proof. Differentiating the equation in the i th -coordinate direction gives (4.17) Using (⋆ ǫ ) then rearranging terms, Proof. Using (4.16) we take the Laplacian of the equation to yield Again using (4.16), we observe (4.20) Substituting this into the result of (4.19) and rearranging terms we arrive at (4.18).
and the result follows.

4.2.
C 0 -estimates. We now turn to the estimates proper, and will frequently take advantage of the special choice of f = ǫf 0 with 0 < ǫ ≤ 1. In particular, we note that f ≤ Cǫ, and where C only depends on the boundary data u 0 , u 1 . In addition, since is linear in t, it follows that f tt = 0. Proof. First we observe that equation (⋆ ǫ ) and the assumption that f > 0 imply that u tt ≤ 0. Using this and the fundamental theorem of calculus yields a lower bound for u depending on inf M u 0 and inf M u 1 .
To obtain an upper bound, let A > 0 (to be chosen later) and definẽ Then by Lemmas 4.3 and 4.4, Proposition 4.9. If u is an admissible solution of (⋆ ǫ ), then there is a constant C (depending on u 0 and u 1 ) such that Proof. Since u tt ≤ 0, it suffices to prove an upper bound for u t at t = 0 and a lower bound for u t at t = 1.
To establish an upper bound at t = 0 we will construct a supersolution to the equation. Let where A, B > 0 are constants to be specified later. Note by the definition of L u , Therefore, by Lemmas 4.3 and 4.4 Since u 0 is admissible, and it follows that Also, In particular, if we choose A > 0 large enough depending on max M f and κ 0 , then L u w ≥ 0, and by the maximum principle w cannot have an interior maximum. Notice w(x, 0) = 0, while Therefore, if we choose B large enough (depending on max M u 1 and min M u 0 ), we can arrange so that w(x, 1) ≤ 0, and consequently and letting t → 0 + we conclude To prove a lower bound for u t at t = 1, we definẽ and the argument proceeds as before.
Proof. If the supremum of ∇u is attained when t = 0 or t = 1 then we are done. Therefore, assume ∇u attains an interior maximum. Let Λ > 0 (to be chosen later) and define We may assume that ∇u is large enough so that ∆u ≤ C.
Proof. Let Q = −∆u + K; then it suffices to prove a bound for sup M Q . Since u is admissible, Q > 0 and so we only need to prove an upper bound for Q. By Lemma 4.6, where Λ > 0 will be chosen. Then If Q attains its maximum when t = 0 or t = 1, then we are done. Therefore, assume Q has an interior maximum, and furthermore that max M ×[0,1] Q > 1. Then at the point where Q attains its maximum, ∇Q = 0 and (4.27) becomes However, if Λ > 0 is chosen larger than min M ×[0,1] ∆f , then we have L u W > 0, a contradiction. Proof. Since u tt < 0, we only need to estimate the infimum of u tt . For the proof we will use the special form of the function f . Recall f = ǫf 0 , where f 0 is given in (4.11). As we observed in (4.23), Therefore, by Lemma 4.7, It follows from the strong maximum principle that u tt cannot have an interior minimum, and the lemma follows. u tt + ∇u t + ∇ 2 u ≤ C.
Proof. Observe that a bound for ∇ 2 u on the boundary is immediate. If we can prove a bound on the 'mixed' term ∇u t , then restricting to equation to t = 0 we have Therefore, a bound on ∇u t implies a bound on u tt . To prove a bound on ∇u t we consider the following auxiliary function Ψ ∶ M × [0, τ ] → R, where 0 < τ < 1 will be chosen later: where A, B > 0 are to be determined.
We claim that by choosing A, B >> 0 large enough and τ > 0 small enough, that Ψ attains a non-positive maximum on the boundary of of M × [0, τ ]. Assuming for the moment this is true, let us see how a bound for ∇u t follows.
Choose a point x 0 ∈ M , and a unit tangent vector Since u t is bounded, an upper bound on ∂ ∂x 1 u t follows. Since X = ∂ ∂x 1 was arbitrary, we obtain a bound on ∇u t (x, 0) .
To see that such a choice of A, B, and τ are possible, we first note that Ψ(x, 0) = 0. (4.29) Since ∇u is bounded, Since u t is also bounded, Therefore, if B is chosen large enough (depending on τ , A, C 1 , and C 2 ), then Assume the maximum of Ψ is attained at a point (x 0 , t 0 ) which is interior (i.e., 0 < t 0 < τ ). Let We can extend η locally via parallel transport along radial geodesics based at x 0 . By construction, By using a cut-off function, we can assume η is globally defined and satisfies (4.34) and the max of H is attained at (x 0 , t 0 ). Therefore, Also, commuting derivatives and using the fact that ∇η(x 0 ) = 0, we have where in the last line we have used (4.32) and the fact that ∇u is bounded. Combining the above, we have (4.37) Also, differentiating the equation in the direction of η, Combining this with (4.37), we get  Then if A > 0 is chosen large enough, the term in braces in (4.39) is bounded below by (A 2)δ 0 . Also, by (4.21), Using the arithmetic-geometric mean inequality, We can assume B is large enough so that δ 0 B > 1. Therefore, Once A is fixed, since f = ǫf 0 , for ǫ > 0 small enough we have Af << 1, hence √ Af > 2Af , which contradicts (4.42). It follows that H (hence Ψ) cannot have an interior maximum.   By Propositions 4.11 and 4.13, the right-hand side is uniformly bounded, and a bound for ∇u t follows.
Next, we give an estimate for the Hessian for solutions to (⋆ ǫ ).
Proof. By Proposition 4.11 the trace of ∇ 2 u is controlled, and therefore it suffices to control either the smallest or largest eignevalue; we will estimate the smallest.
Since ∇ 2 u is controlled on the boundary M × {0, 1}, we only need an interior estimate. To this end, let λ min (x, t) = min gx(X,X)=1 ∇ 2 u(x, t)(X, X) (4.44) denote the smallest eigenvalue of ∇ 2 u at (x, t). Define where A, B > 0 will be specified later, but will only depend on the boundary data u 0 , u 1 . If H attains its minimum on the boundary M × {0, 1}, then we are done. Therefore, assume H attains its minimum at a point (x 0 , t 0 ), with 0 < t 0 < 1. Let η ∈ T x 0 M be a unit eigenvector corresponding to the smallest eigenvalue λ min (x 0 , t 0 ). As in the proof of Proposition 4.13, we extend η locally by parallel transport along radial geodesics originating at x 0 , hence η satisfies (4.32). By introducing a cutoff, we can assume that η is globally defined (in space), with η ≤ 1 on M and η = 1 near x 0 . Now let Since η 2 = 1 in a neighborhood of x 0 , it follows that for x near x 0 , with equality when (x, t) = (x 0 , t 0 ). It follows that Φ attains a local minimum at (x 0 , t 0 ), hence (4.48) By commuting derivatives and using (4.32), we also have (at Since ∇u and ∆u are bounded, it follows   By (4.51), the cross terms above can be written Also, commuting derivatives in the second-to-last term in (4.52) gives Substituting (4.53) and (4.54) into (4.52), pairing with η i η j , and rearranging terms gives Notice the first three terms in (4.55) correspond to the leading terms of (4.50). Therefore, substituting gives (4.56) Let Then (4.56) implies (4.57) Note that using the equation, the first line in (4.57) can be written where in the final inequality we used the fact that f = ǫf 0 with f 0 > 0 depending only on the boundary data for u.
The second line in (4.57) can be estimated as where again we used the special form of f . Combining (4.57), (4.58), and (4.59) we conclude Next, we recall the identity (4.17): Writing the equation as in (4.43), we can estimate the first term on the second line above by Since ∆u is bounded, it follows that Using the bound for the gradient, the last term in the second line of (4.61) can easily be estimated as −2⟨∇u, ∇f ⟩ ≥ −C ∇f ≥ −Cf.
Combining this with (4.60) gives If ∇ 2 u is large enough, we may assume that and once ∇ 2 u is large enough, we conclude L u Ψ < 0. As a consequence, either ∇ 2 u is bounded at an interior minimum of Ψ, or else Ψ attains its minimum when t = 0 or t = 1. In either case we see that Ψ is bounded from below, hence ∇ 2 u is bounded.
We claim that I is closed: let {u i = u ǫ i } be a sequence of admissible solutions with ǫ i ≥ ǫ 0 > 0. The preceding a priori estimates imply there is a constant C (independent of ǫ) such that To apply Evans/Krylov and obtain Hölder estimates for the second derivatives, we need to verify the concavity of the operator. To do so we will rewrite the operator as We view M as a function of the space-time Hessian of u, and to simplify the calculations let us fix a point x 0 ∈ M and introduce normal coordinates {x 1 , x 2 } based at x 0 . We also denote derivatives with respect to t with the subscript 0; then at x 0 To simplify just write f (x 0 ) = f and K(x 0 ) = K. Expressed in this way it is easy to see that the equation remains elliptic: differentiating with respect to the entries of the space-time Hessian {r ab }, 0 ≤ a, b ≤ 2, we have Differentiating again, M 00,00 = 0 Then M ab,cd η ab η cd = 4 (−r 11 − r 22 + K) 2 r 01 η 01 + r 02 η 02 η 11 + η 22 (4.75) The first term above can be estimated by 4 (−r 11 − r 22 + K) 2 r 01 η 01 + r 02 η 02 η 11 + η 22 ≥ − 2(r 2 01 + r 2 02 + f ) (−r 11 − r 22 + K) 3 η 11 + η 22 2 − 2 (−r 11 − r 22 + K) r 01 η 01 + r 02 η 02 2 (r 2 01 + r 2 02 + f ) .
Substituting into (4.75) gives  It follows that M is a convex function of the space-time Hessian, and applying Evans-Krylov [11] [15] we conclude there is a constant C = C(ǫ) such that Applying the Schauder estimates we obtain bounds on derivatives of all orders, and it follows that the set I is closed.
To verify that I is open, it suffices to study the linearized equation; i.e., given ψ ∈ C ∞ (M ×[0, 1]), we need to solve L u i ϕ = ψ with ϕ satisfying Dirichlet boundary conditions. The solvability of this linear problem follows from [12], Theorem 6.13.
Since I is open, closed, and non-empty, it follows that I = [ǫ 0 , 1]. This proves the existence of solutions. Uniqueness will follow from the following comparison lemma: Lemma 4.18. Suppose u,ũ ∈ C ∞ are admissible and satisfy where f 1 ≤ f 2 . Assume further that on the boundary, Then w 0 = u and w 1 =ũ. By (4.77), Sinceũ − u = 0 on the boundary, by the maximum principle we conclude thatũ − u ≤ 0 on M × [0, 1].
If u,ũ are solutions of (⋆ ǫ ) with the same boundary conditions, then we may apply the preceding comparison lemma with f 1 = f 2 = ǫf 0 and conclude that u = v. This completes the proof of Proposition 4.17.

Metric space structure
In this section we use the existence of regularizable geodesics to define a metric space structure on Γ + 1 . First we show that our definition does indeed define a metric space, and then we establish nonpositivity of curvature in the sense of Alexandrov. For simplicity we focus entirely on the positive cone setting, the proofs for the negative cone being directly analogous. 5.1. Metric space structure. Definition 5.1. Given u 0 , u 1 ∈ Γ + 1 , let d(u 0 , u 1 ) denote the distance of the unique regularizable geodesic connecting u 0 to u 1 .
We will establish that d does indeed define a metric space structure in what follows. First we establish nondegeneracy of the distance, for which we require a preliminary lemma. Then we establish the triangle inequality, finishing the proof.
Lemma 5.2. Let u be a regularizable geodesic. Then u has constant energy density.
Proof. Fixing a regularization we directly compute using the uniform C 1,1 bounds to yield d dt Taking ǫ to zero and using the convergence properties finishes the lemma.
Proposition 5.3. Given u 0 , u 1 ∈ Γ + 1 and u the unique regularizable geodesic connecting u 0 to u 1 , one has Proof. Observe that the geodesic equation implies u tt ≤ 0, and so we obtain the pointwise inequality Thus using Hölder's inequality and the Gauss-Bonnet theorem we have A similar argument yields Since geodesics automatically have constant energy density by Lemma 5.2, the result follows.
Our next goal is to establish the triangle inequality for the metric d. To do this we require the existence of approximate geodesics connecting paths in Γ + 1 . The proof of the following proposition is a straightforward adaptation of the proof of Proposition 4.17  In particular, one has Proof. It suffices to consder the case when v = 0 by changing basepoints. We apply Proposition 5.4 in the case that γ 1 (s) = 0, γ 2 (s) = u, obtaining the two-paramter family Υ. Let L(s, ǫ) denote the length of Υ(⋅, s, ǫ), and let λ(s) denote the length of the given curve u up to time s, i.e.
The main step is to obtain a lower bound for L(s, ǫ) + λ(s). To do this we use the ǫ-approximations and compute Using the convergence properties and sending ǫ to zero yields the result.
Proof. This follows immediately from Propositions 5.3 and 5.5.

5.2.
Nonpositive curvature. Next we establish the nonpositive curvature of the metric spaces (Γ ± 1 , d) in the sense of Alexandrov. First let's record some notation and recall the definition of nonpositive curvature. Given A, B ∈ Γ + 1 , let AB(s) denote γ(s), where γ ∶ [0, 1] → Γ + 1 is the unique regularizable C 1,1 geodesic connecting A to B. Note that a priori AB(s) is only C 1,1 , but this suffices for our purposes in establishing metric space properties. The condition of nonpositive curvature then amounts to the claim that for all A, B, C one has the inequality: The first step in establishing this is to show that Jacobi fields are convex along geodesics. Proof. To simplify notation we set X = ∂Υ ∂t , Y = ∂Υ ∂s . Then we compute 1 2 Using this it follows that Rearranging this yields as required.
Proposition 5.8. The metric spaces (Γ ± 1 , d) are nonpositively curved in the sense of Alexandrov. Proof. Fix three points A, B, C ∈ Γ + 1 , let γ 1 (s) = A for all s, and let γ 2 (s) be the an ǫ-approximate geodesic B to C. Let Υ denote the family associated to these two paths guaranteed by Proposition 5.4. Treating ǫ as small but fixed, let E(s) denote the total energy of the path Υ(x, t, s, ǫ) connecting A to γ 2 (s). We aim to show that E(s) is convex up to an error of order ǫ. We simplify notation and set X = ∂Υ ∂t , Y = ∂Υ ∂s . Then we compute 1 2 Next we compute, using Lemma 5.7,

It follows directly that
Note that for each s the energies E(s) converge as ǫ → 0 to d(A, BC(s)) 2 , thus we conclude by sending ǫ to zero in the above inequality that as required.

Functional determinant and the inverse Gauss curvature flow
In [21], Polyakov proved a remarkable formula for the ratio of regularized determinants of two conformal metrics. Given conformal metrics g 0 and g u = e 2u g on a surface M , The integral in this formula is often referred to as the Liouville energy, and we will denote it by J: Since J is not scale-invariant, it is convenient to also consider the normalized version of J, which we denote by S: and we denote the (normalized) version by F ∶ W 1,2 → R A first variation calculation of J at u is and a first variation of F is We want to consider the (negative) gradient flow for F with respect to the metric we have defined for Γ + 1 . In view of (6.5) and (2.3), It follows that the (negative) gradient flow for F is given by or written in metric terms, We will refer to (6.7) and (6.8) as the inverse Gauss curvature flow, or IGCF. For the cone of metrics with negative curvature, the flow is defined by 6.1. Formal Properties. In this section we establish the geodesic convexity of the Liouville energy, the convexity of the normalized Liouville energy along flow lines as well as the monotonicity of distances along flow lines. Remarkably, all three properties rely on a sharp application of a curvature-weighted Poincare inequality [1]. We include the short proof as this result seems to not be well-known. Proposition 6.1. (Andrews [1], cf. [8] pg. 517) Let (M n , g) be a closed Riemannian manifold with positive Ricci curvature. Given φ ∈ C ∞ (M ) such that ∫ M φdV = 0, then with equality if and only if φ ≡ 0 or (M n , g) is isometric to the round sphere.
Proof. Since ∫ M φ = 0 there exists ψ such that ∆ψ = φ. Observe that Proof. We directly compute Proposition 6.7. Let u be a solution to (6.11). Then for 0 ≤ t < T , Proof. We apply the maximum principle to the result of Lemma 6.6. At a minimum point, certainly − K K 2 ∆K ≥ 0 (since K < 0), ∇K = 0, and K < K. It follows that the minimum of K is nondecreasing along the flow. A similar argument shows that the maximum is nonincreasing. Proposition 6.8. Let u be a solution to (6.11). There exists a constant C = C(M, u 0 ) such that for all 0 ≤ t < T , Proof. We apply the maximum principle directly to (6.11). At a spacetime minimum point for u, we have ∆ g 0 u ≥ 0, and hence Similarly, at a spacetime maximum one has ∆ g 0 u ≤ 0 and hence where we applied the Cauchy-Schwarz inequality in the final line. Next, we commute derivatives to yield Proof. Fix some constant A and consider the function ∇ 2 0 u(X, X) X 2 This is continuous, and an upper bound for β implies an upper bound for the Hessian of u. Fix a constant A and consider the function Φ(x, t) = tβ + A ∇u 2 0 . We claim that for A chosen sufficiently large that there is an a priori bound for an interior maximum on [0, 1]. Suppose some point (x, t) is such an interior spacetime maximum for Φ. Fix a unit vector Z ∈ T x M realizing the supremum in the definition of β(x). We may extend Z in a neighborhood of x by parallel transport along radial geodesics. This yields We may extend Z to all of M by multiplying by a cutoff function. We observe that the function Ψ(x, t) = t∇ 2 0 (Z, Z) + A ∇u 2 0 also has a spacetime maximum at (x, t). Combining Lemmas 6.9 and 6.10 yields that, at x, where the last line follows by choosing A sufficiently large with respect to Λ and using the a priori gradient bound. This implies an a priori upper bound for the Hessian of u at (x, t), and hence since t ≤ 1 an a priori bound for Ψ at (x, t). This yields an a priori upper bound for ∇ 0 Z ∇ 0 Z u for any interior time t > ǫ. Combined with some ineffective estimate depending on the given solution for [0, ǫ) yields an a priori upper bound on [0, T ). Since the Laplacian of u is uniformly bounded below, this then yields the full a priori Hessian bound. Proposition 6.12. Let (M 2 , g 0 ) be a compact Riemann surface such that K 0 < 0. Given u ∈ Γ − 1 , the solution to (6.11) exists for all time and converges exponentially to a metric of constant negative scalar curvature.
Proof. Propositions 6.8 and 6.7 guarantee uniform estimates on u, K and 1 K . Proposition 6.11 then implies a uniform estimate for the Hessian of u. We now observe that the operator Φ(u) = K−K K is convex, and with the uniform upper and lower bounds on curvature, is uniformly elliptic. Thus the Evans-Krylov theorem [11], [15] yields an a priori C 2,α estimate for u. Schauder estimates can then be applied to obtain estimates of every C k,α norm of u. It follows that the solution exists on [0, ∞) and every sequence of times approaching infinity admits a subsequence such that {u t i } converges to a limiting function u ∞ . Using that the flow is the gradient flow for F it follows easily that the limiting metric u ∞ has constant curvature, and since this metric is unique the whole flow converges to u ∞ . 6.3. The sphere. We now consider the case of K 0 > 0, and so M ≅ S 2 . In this case we are studying the flow ∂ ∂t u = K − K K . (6.13) 6.3.1. Evolution Equations. To begin we build up some evolution equations. First we rewrite the evolution of u in terms of the linearized operator. Lemma 6.13. Given u a solution to (6.13) we have Hence using (6.13) we have Lemma 6.14. Let u be a solution to (6.13). Then Proof. We directly compute Proof. We observe that at a spacetime minimum for u, one has ∆ g 0 u ≥ 0, and hence The result follows from the maximum principle. Proposition 6.20. Let u be a solution to (6.13). There exists a constant C = C(u 0 ) such that for all smooth existence times t of the flow one has ∇u(t) 2 L 2 ≤ C [1 + t] Proof. We use Lemma 6.17 with the estimate of Proposition 6.19 to yield K t dV t < 2π.
Using Proposition 6.18 we estimate where the last inequality follows by choosing R small with respect to C. Invoking [26] Theorem 3.2 we conclude a uniform H 2 2 bound for u, which by the Sobolev inequality implies the uniform bound for u. Proposition 6.22. Let u be a solution to (6.13) on [0, T ) such that u ≤ Λ. There exists a constant C = C(T, u 0 ) such that Proof. Let Φ = tK −1 + Au, where A > 0 is a constant yet to be determined. Using Lemmas 6.13, 6.15 and Proposition 6.21 we have . where the constant δ > 0 is determined by the upper bound for u. If we choose A sufficiently large with respect to δ, then at a sufficiently large maximum for K −1 we obtain The result follows from the maximum principle.
Proposition 6.23. Let (M 2 , g 0 ) be a compact Riemann surface such that K 0 > 0. Given u ∈ C ∞ (M ), the solution to (6.13) exists for all time.
Proof. Combining Propositions 6.21, 6.18, and 6.22, we obtain uniform estimates on u, K, and K −1 for any finite existence time T . Given these the higher order estimates follow as in the proof of Proposition 6.11 and Theorem 6.12, and so the long time existence follows.
Proposition 6.24. Given (S 2 , g S 2 ), and u ∈ Γ + 1 , the solution to (6.13) exists for all time and converges weakly in the distance topology to a minimizer for F in the completion (Γ + 1 , d).
Proof. We provide a sketch of the proof, as the key ingredients have been established already and the result follows formally from prior results. The principal tool we require is a general result [2] concerning convergence of weak gradient flows in metric spaces.
As we have established that (Γ + 1 , d) is NPC and convex by unique regularizable C 1,1 geodesics, the argument of ([24] Lemma 5.9) shows that the completion (Γ + 1 , d) is also an NPC space. Moreover, let F denote the extension of F to Γ + 1 via its canonical lower-semicontinuous extension. Following the argument of ([24] Lemma 5.15), it follows that F is geodesically convex. Moreover, we know that the minimum of F is attained by any constant curvature metric, which exists by assumption. It follows that these also realize the minimum of F . Hence we have verified the setup of ([18] Theorem 1.13), guaranteeing the existence of a global weak solution to the gradient flow of F with arbitrary initial data. Moreover, following the argument of ([25] Theorem 1.1) we can verify that the smooth global solutions of Proposition 6.23 coincide with the weak solutions constructed via ([18] Theorem 1.13). The convergence of the weak flows, and hence the smooth flows, to a minimizer for F in the weak distance topology now follows from [2] Theorem 1.5.