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Cheeger constants of surfaces and isoperimetric inequalities. (English) Zbl 1183.53030
If \((M^n,g)\) is a Riemannian manifold of infinite volume, the isoperimetric profile function of \(M^n\) is the function \(I_{M}:\mathbb R^+\rightarrow\mathbb R^+\) defined by
\[ I_{M}(t)=\inf_\Omega \{\text{vol}_{n-1}(\partial\Omega): \Omega\subset M^n, \text{ vol}_n(\Omega)=t\}, \]
where \(\Omega\) ranges over all regions of \(M^n\) with smooth boundary. One can similarly define an isoperimetric profile function \(I_M:\mathbb{N}\rightarrow\mathbb N\) for simplicial manifolds \(M^n\). The main result of this paper is the following: Let \(S\) be a plane with holes equipped either with a Riemannian metric or with a simplicial complex structure. Assume that there is some \(K>0\) such that for all \(t\in[K,100K],\) \(I_S(t)\geq10^2\sqrt{t}\). Then for all \(t>K\), \(I_S(t)\geq\frac{1}{\sqrt{K}}t\).
As a corollary one obtains a similar result for finite genus surfaces. Notice that the result above generalizes a similar result of Gromov for surfaces \((S,g)\) which are homeomorphic to the plane.
The author uses bounds on the Cheeger constants to study isoperimetric profiles of surfaces. If \((M^n,g)\) is a Riemannian manifold, one defines the Cheeger constant \(h\) of \(M\) by
\[ h(M)=\inf_A \bigg\{\frac{\text{vol}_{n-1}(\partial A)}{\text{vol}_n(A)} \bigg\}: \text{vol}_n(A)\leq\frac{1}{2} \text{vol}_n(M)\}, \]
where \(A\) ranges over all open subsets of \(M\) with smooth boundary. If \(M\) is a simplicial manifold, one can similarly define the Cheeger constant of \(M\). The author gives new proofs of the Cheeger constant bounds which have the advantage that they are more direct than the existing ones. More concretely, it is proved the following: Let \(S\) be a closed orientable surface of genus \(g\geq1\) (resp. \(S\) of genus 0) equipped either with a Riemannian metric or with a simplicial complex structure. Let \(A(S)\) be its (Riemannian or simplicial) area. Then, the Cheeger constant, \(h(S)\), of \(S\) satisfies the inequality \(h(S)\leq\frac{4\cdot10^3\cdot g^2}{\sqrt{A(S)}}\) (resp. \(\frac{16}{\sqrt{A(S)}})\).
An interesting question asked by the author is whether Gromov’s theorem on filling area has an analogue for higher dimensional filling functions \(FV_i\), \(i\geq2\), which are defined in the paper. The bounds on the Cheeger constants of surfaces can be used to obtain some partial result in this direction. There is also a conjecture stated in the paper: If \(FV_i\) is Euclidean for \(i=2,\dots,k-1\) and \(FV_k\) is sub-Euclidean, then \(FV_k\) is linear. A theorem proved by the author in the paper gives some evidence in favor of this conjecture.

MSC:
53C20 Global Riemannian geometry, including pinching
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
20F65 Geometric group theory
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