On the minimization of total mean curvature

TL;DR

This paper explores extensions of Minkowski's inequality relating mean curvature and surface area, proving its validity for certain axisymmetric convex domains and establishing bounds involving absolute mean curvature.

Contribution

It extends Minkowski's inequality to axisymmetric convex domains and shows the inequality's limitations in more general cases.

Findings

Minkowski's inequality holds for axisymmetric convex domains.

The inequality does not generally hold beyond axisymmetric convex cases.

The inequality with absolute mean curvature remains valid for all axisymmetric domains.

Abstract

In this paper we are interested in possible extensions of an inequality due to Minkowski: $\int_{\partial\Omega} H\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ valid for any regular open set $\Omega\subset\mathbb{R}^3$, where $H$ denotes the scalar mean curvature and $A$ the area. We prove that this inequality holds true for axisymmetric domains which are convex in the direction orthogonal to the axis of symmetry. We also show that this inequality cannot be true in more general situations. However we prove that $\int_{\partial\Omega} |H|\,dA \geq \sqrt{4\pi A(\partial\Omega)}$ remains true for any axisymmetric domain.