Talk

Optimal isoperimetric inequalities for 2-dimensional surfaces in Hadamard-Cartan manifolds in any codimension

  • Felix Schulze (University College London)
A3 01 (Sophus-Lie room)

Abstract

Let (Mn,g) be simply connected, complete, with non-positive sectional curvatures, and 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 S=Σ. We use a weak mean curvature flow, obtained via elliptic regularisation, starting from Σ, to show that S satisfies the optimal Euclidean isoperimetric inequality: 6πM[S](M[Σ])3/2. We also obtain an optimal estimate in case the sectional curvatures of M are bounded from above by 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 L2.