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

Abstract

AbstractLet $$(M^n,g)$$(Mn,g) be simply connected, complete, with non-positive sectional curvatures, and $$\Sigma $$Σ 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 $$\partial S = \Sigma $$∂S=Σ. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from $$\Sigma $$Σ, to show that S satisfies the optimal Euclidean isoperimetric inequality: $$ 6 \sqrt{\pi }\, \mathbf {M}[S] \le (\mathbf {M}[\Sigma ])^{3/2} $$6πM[S]≤(M[Σ])3/2. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by $$-\kappa < 0$$-κ<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 $$L^2$$L2.