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

2018/02/01 by Felix Schulze, Schulze, Felix · 2 citations
Mathematics · #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematics and Applications #Point processes and geometric inequalities

paper · pdf · doi:10.48550/arxiv.1802.00226

openalex publication_date 2018/02/01 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

Let (Mⁿ,g) be simply connected, complete, with non-positive sectional\ncurvatures, and \Σ a 2-dimensional closed integral current (or flat chain\nmod 2) with compact support in M. Let S be an area minimising integral\n3-current (resp. flat chain mod 2) such that \∂ S = \Σ. We use a\nweak mean curvature flow, obtained via elliptic regularisation, starting from\n\Σ, to show that S satisfies the optimal Euclidean isoperimetric\ninequality: 6 \√(\π) , \M[S] \≤ (\M[\Σ])^3/2 .\nWe also obtain an optimal estimate in case the sectional curvatures of M are\nbounded from above by -\κ < 0 and characterise the case of equality. The\nproof follows from an almost monotonicity of a suitable isoperimetric\ndifference along the approximating flows in one dimension higher and an optimal\nestimate for the Willmore energy of a 2-dimensional integral varifold with\nfirst variation summable in L².\n

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• Green functions and a positive mass theorem for asymptotically hyperbolic 3-manifolds
• Isoperimetric inequalities in Hadamard spaces of asymptotic rank two

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