Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds

TL;DR

This paper proves optimal isoperimetric inequalities for surfaces in Cartan-Hadamard manifolds using a weak mean curvature flow, extending classical results to higher codimensions and curved spaces.

Contribution

It introduces a novel application of elliptic regularisation-based mean curvature flow to establish sharp isoperimetric inequalities in non-positively curved manifolds.

Findings

Proves Euclidean isoperimetric inequality in Cartan-Hadamard manifolds.

Establishes optimal estimates under negative curvature bounds.

Characterizes cases of equality in the inequalities.

Abstract

Let $(M^n,g)$ be simply connected, complete, with non-positive sectional curvatures, and $\Sigma$ a 2-dimensional closed integral current (or flat chain mod 2) with compact support in $M$. Let $S$ be an area minimising integral 3-current (resp. flat chain mod 2) such that $\partial S = \Sigma$. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $\Sigma$, to show that S satisfies the optimal Euclidean isoperimetric inequality: $ 6 \sqrt{\pi}\, \mathbf{M}[S] \leq (\mathbf{M}[\Sigma])^{3/2} $. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by $-\kappa < 0$ and characterise the case of equality. The proof follows from an almost monotonicity of a suitable isoperimetric difference along the approximating flows in one dimension higher and an optimal estimate for the Willmore energy of a 2-dimensional integral varifold with first variation summable in $L^2$.