Minkowski inequality in Cartan-Hadamard manifolds

TL;DR

This paper proves a sharp Minkowski inequality for convex surfaces in nonpositively curved 3-manifolds using harmonic mean curvature flow, leading to improved geometric bounds and isoperimetric inequalities.

Contribution

It introduces a new Minkowski inequality in Cartan-Hadamard manifolds and connects it to isoperimetric inequalities via harmonic mean curvature flow.

Findings

Established a sharp lower bound for total mean curvature in Cartan-Hadamard 3-manifolds.

Extended the Minkowski inequality to higher-dimensional Cartan-Hadamard manifolds.

Derived a Bonnesen-style isoperimetric inequality for convex-distance surfaces.

Abstract

Using harmonic mean curvature flow, we establish a sharp Minkowski type lower bound for total mean curvature of convex surfaces with a given area in Cartan-Hadamard 3-manifolds. This inequality also improves the known estimates for total mean curvature in hyperbolic 3-space. As an application, we obtain a Bonnesen-style isoperimetric inequality for surfaces with convex distance function in nonpositively curved 3-spaces, via monotonicity results for total mean curvature. This connection between the Minkowski and isoperimetric inequalities is extended to Cartan-Hadamard manifolds of any dimension.