Cheeger constants of surfaces and isoperimetric inequalities

We show that the Cheeger constant of compact surfaces is bounded by a function of the area. We apply this to isoperimetric profiles of bounded genus non-compact surfaces, to show that if their isoperimetric profile grows faster than $\sqrt t$, then it grows at least as fast as a linear function. This generalizes a result of Gromov for simply connected surfaces. We study the isoperimetric problem in dimension 3. We show that if the filling volume function in dimension 2 is Euclidean, while in dimension 3 is sub-Euclidean and there is a $g$ such that minimizers in dimension 3 have genus at most $g$, then the filling function in dimension 3 is `almost’ linear.

💡 Research Summary

The paper consists of three tightly linked parts, each addressing how geometric isoperimetric quantities are controlled by topological complexity.
In the first part the authors consider a compact two‑dimensional Riemannian manifold (M) (a closed surface) and study its Cheeger constant
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