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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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