A quantitative obstruction to collapsing surfaces

We provide a quantitative obstruction to collapsing surfaces of genus at least 2 under a lower curvature bound and an upper diameter bound.


Introduction
S. Alesker posed the following question at MathOver ow [1]. Let (M i ) be a sequence of 2-dimensional orientable closed surfaces of genus g ≥ endowed with smooth Riemannian metrics of Gaussian curvature at least − and diameter at most D. By the Gromov compactness theorem, one can choose a subsequence converging in the Gromov-Hausdor (GH) sense to a compact Alexandrov space with curvature at least − and Hausdor dimension , , or . Let us assume that the limit space has dimension . Then it is either a circle or a segment. Can these possibilities (circle and segment) be obtained in the limit M of (M i )? We show that these possibilities cannot occur, and quantify this statement by providing an explicit lower bound for the lling radius of M. For related results see [2].

Impossibility of collapse
We prove the impossibility of collapse in dimension 2, in the following sense.
Theorem 2.1. The distance between a strongly isometric map from a closed orientable surface M of genus g ≥ of Gaussian curvature K ≥ − and diameter at most D to a metric space Z, and a map from M to a graph in Z, is at least π(g− ) sinh D .
Thus we obtain a quantitative lower bound rather than merely the nonexistence of Shioya-Yamaguchi-type collapse to spaces of positive codimension (see [3,4] Recall that the systole of a Riemannian manifold M is the least length of a noncontractible loop of M. For an overview of systolic geometry see [5]. The lling radius FillRad M of a closed n-dimensional manifold M is de ned as the in mum of all ϵ > such that the inclusion of M in its ϵ-neighborhood in any strongly isometric embedding of M in a Banach space sends the fundamental homology class [M] of M to the zero class, by means of the induced homomorphism on Hn(M). Here the embedding can be taken to be into the space of bounded functions on M which sends a point p ∈ M to the distance function from p. This embedding is strongly isometric (ambient distance restricted to M coincides with intrinsic distance on M) if the function space is equipped with the sup-norm. Proof. Consider a strongly isometric embedding of the surface M into a Banach space B. The space B can be assumed nite-dimensional if the metric condition is relaxed to a requirement of being bilipschitz with to a bilipschitz factor arbitrarily close to ; see [6]. Suppose M is " lled" (in the homological sense) by a chain C (in the sense that M is the boundary of C). Then the induced homomorphism Hn(M) → Hn(C) sends [M] to the zero class. Consider a triangulation of C into in nitesimal simplices (here the term "in nitesimal" is used informally in its meaning "su ciently small" though this could be rendered rigorous as in [7]).
We argue by contradiction. Let R > be strictly smaller than a sixth of the systole. Suppose the chain C is contained in an open R-neighborhood of M in B. We will retract C back to M, while xing the subset M ⊆ C, contradicting the fact that the nonvanishing fundamental class [M] is sent to a zero class in C.
For each vertex of the triangulation of C, we choose a nearest point of M. To extend the retraction to theskeleton of C, we map each edge (of a triangle of the triangulation) to a minimizing path joining the images of the two vertices in M. The length of such a minimizing path is less than R (plus the in nitesimal sidelength of the triangle) by the triangle inequality. Hence the boundary of each -cell of the triangulation is sent to a loop of length at most R (plus an in nitesimal). Since this length is less than the systole of M, the map can now be extended to the -skeleton of C.
To extend the map to the skeleton, note that the universal cover of M is contractible and hence π (M) = , and similarly for the higher homotopy groups. Therefore the skeletal retraction extends to all of C inductively. The contradiction completes the proof of the lemma.
Proof of Theorem 2.1. We exploit Gromov's notion of the lling radius of a manifold [8]. The argument relies only on basic Jacobi eld estimates and basic homotopy theory. We seek a suitable lower bound so as to rule out positive-codimension collapse. Choose a noncontractible closed geodesic γ ⊆ M of length equal to the systole sys(M). Consider the normal exponential map along γ. Using the lower curvature bound, we obtain an upper bound on the total area of M as sys(M) sinh(D), where D is the diameter. The bound follows by applying Rauch bounds on Jacobi elds (this is an ingredient in the proof of Toponogov's theorem); see e.g., Cheeger-Ebin [9,Theorem 5.8,. The bound results from comparison with the area of a hyperbolic collar of width D around a closed geodesic of the same length as γ. Therefore, the systole is bounded below as follows: Meanwhile the area is bounded below by the Gauss-Bonnet theorem: where g is the genus. Furthermore the lling radius of M is bounded below by a sixth of the systole by Gromov's Lemma 2.3. Therefore the bound (2.1) implies The theorem now follows from the fact the distance between a strongly isometric map from M to a metric space Z and a map from M to a graph in Z is bounded below by the lling radius; see e.g., [8,p. 127,Example]. This proves that aspherical surfaces of curvature bounded below by − with diameter bounded above by D cannot collapse, so that a GH limit is necessarily 2-dimensional as follows.
To prove Corollary 2.2, note that if a metric on M is su ciently close to a nite graph Γ in the sense of the GH distance, then the construction of the proof of Lemma 2.3 produces a map from M to Γ which is close to the embedding of M in Z, contradicting the lower bound (2.2).