A quantitative obstruction to collapsing surfaces

Mikhail G. Katz

doi:10.1515/math-2019-0103

Article Open Access

A quantitative obstruction to collapsing surfaces

Mikhail G. Katz

Published/Copyright: October 18, 2019

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$Open Mathematics$

From the journal Open Mathematics Volume 17 Issue 1

Abstract

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

Keywords: curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance

1 Introduction

S. Alesker posed the following question at MathOverflow [1]. Let (M_i) be a sequence of 2-dimensional orientable closed surfaces of genus g ≥ 2 endowed with smooth Riemannian metrics of Gaussian curvature at least –1 and diameter at most D. By the Gromov compactness theorem, one can choose a subsequence converging in the Gromov-Hausdorff (GH) sense to a compact Alexandrov space with curvature at least –1 and Hausdorff dimension 0, 1, or 2. Let us assume that the limit space has dimension 1. 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 filling radius of M. For related results see [2].

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 ≥ 2 of Gaussian curvature K ≥ –1 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−1)3sinh⁡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]).

Corollary 2.2

Let D > 0. GH limits of metrics on a closed orientable surface of genus g ≥ 2 with Gaussian curvature at least –1 and diameter at most D are necessarily 2-dimensional.

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 filling radius FillRad M of a closed n-dimensional manifold M is defined as the infimum of all ϵ > 0 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 H_n(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.

Lemma 2.3

(Gromov’s lemma) The systole of an aspherical manifold M is at most six times the filling radius of M.

Proof

Consider a strongly isometric embedding of the surface M into a Banach space B. The space B can be assumed finite-dimensional if the metric condition is relaxed to a requirement of being bilipschitz with to a bilipschitz factor arbitrarily close to 1; see [6]. Suppose M is “filled” (in the homological sense) by a chain C (in the sense that M is the boundary of C). Then the induced homomorphism H_n(M) → H_n(C) sends [M] to the zero class. Consider a triangulation of C into infinitesimal simplices (here the term “infinitesimal” is used informally in its meaning “sufficiently small” though this could be rendered rigorous as in [7]).

We argue by contradiction. Let R > 0 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 fixing 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 the 1-skeleton 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 2R (plus the infinitesimal sidelength of the triangle) by the triangle inequality. Hence the boundary of each 2-cell of the triangulation is sent to a loop of length at most 6R (plus an infinitesimal). Since this length is less than the systole of M, the map can now be extended to the 2-skeleton of C.

To extend the map to the 3 skeleton, note that the universal cover of M is contractible and hence π₂(M) = 0, 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 filling radius of a manifold [8]. The argument relies only on basic Jacobi field 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 2 sys(M) sinh(D), where D is the diameter. The bound follows by applying Rauch bounds on Jacobi fields (this is an ingredient in the proof of Toponogov’s theorem); see e.g., Cheeger-Ebin [9, Theorem 5.8, pp. 97–98]. 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:

sys(M)≥area(M)2sinh⁡D. (2.1)

Meanwhile the area is bounded below by the Gauss-Bonnet theorem:

area(M)≥−∫MK=2π(2g−2),

where g is the genus. Furthermore the filling radius of M is bounded below by a sixth of the systole by Gromov’s Lemma 2.3. Therefore the bound (2.1) implies

FillRad(M)≥16sys(M)≥area(M)12sinh⁡D≥π(g−1)3sinh⁡D. (2.2)

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 filling radius; see e.g., [8, p. 127, Example]. This proves that aspherical surfaces of curvature bounded below by –1 with diameter bounded above by D cannot collapse, so that a GH limit is necessarily 2-dimensional. □

To prove Corollary 2.2, note that if a metric on M is sufficiently close to a finite 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).

References

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Received: 2019-05-20

Accepted: 2019-09-23

Published Online: 2019-10-18

This work is licensed under the Creative Commons Attribution 4.0 Public License.

Articles in the same Issue

https://doi.org/10.1515/math-2019-0103

Keywords for this article

curvature; diameter; volume; filling radius; systole; Gromov-Hausdorff distance

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