Wulff inequality for minimal submanifolds in Euclidean space

TL;DR

This paper establishes a sharp Wulff inequality for minimal submanifolds with boundary in Euclidean space, incorporating anisotropic weights and revealing independence of the inequality constant from these weights.

Contribution

It proves a new Wulff inequality for minimal submanifolds with boundary, with a constant depending only on dimensions and not on anisotropic weights.

Findings

The inequality is sharp for certain cases when m=1,2.

The inequality constant is independent of the anisotropic weights.

The result generalizes classical inequalities to anisotropic minimal submanifolds.

Abstract

In this paper, we prove a Wulff inequality for $n$-dimensional minimal submanifolds with boundary in $\mathbb{R}^{n+m}$, where we associate a nonnegative anisotropic weight $\Phi: S^{n+m-1}\to \mathbb{R}^{+}$ to the boundary of minimal submanifolds. The Wulff inequality constant depends only on $m$ and $n$, and is independent of the weights. The inequality is sharp if $m=1, 2$ and $\Phi$ is the support function of ellipsoids or certain type of centrally symmetric long convex bodies.