Schulze, F; (2020) Optimal isoperimetric inequalities for surfaces in any codimension in Cartan-Hadamard manifolds. Geometric and Functional Analysis , 30 pp. 255-288. 10.1007/s00039-020-00522-8.

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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: 6π−−√M[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.

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