Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds

Mohammad Ghomi

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arXiv · 2608.12020

Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds

Mohammad Ghomi

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Abstract

We establish a sharp lower bound for the total Gauss-Kronecker curvature of convex hypersurfaces in Cartan-Hadamard manifolds with pinched negative curvature. The bound holds in all dimensions when the diameter is small relative to the curvature scale, and in dimensions 4 and 5 without any restriction on the diameter, provided that the pinching is sufficiently tight. The proofs are based on the Chern-Gauss-Bonnet theorem and weighted Hsiung-Minkowski inequalities. As an application, we obtain the isoperimetric inequality of the Cartan-Hadamard conjecture in dimension 5 under sufficiently pinched curvature.

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Mohammad Ghomi. 2026-08-12. Total curvature and isoperimetric inequalities in pinched Cartan-Hadamard manifolds. https://arxiv.org/abs/2608.12020

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