Cheeger constants of surfaces and isoperimetric inequalities

Abstract

We show that if the isoperimetric profile of a bounded genus non-compact surface 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 it 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.