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.