The Maximum Principle for Minimal Varieties of Arbitrary Codimension

TL;DR

This paper establishes a maximum principle for minimal varieties of any codimension, showing they cannot touch boundaries with positive sum of the smallest principal curvatures, extending to bounded mean curvature cases.

Contribution

It generalizes the maximum principle to arbitrary codimension minimal varieties and varieties with bounded mean curvature in Riemannian manifolds.

Findings

Minimal varieties cannot touch boundary points with positive sum of smallest principal curvatures.

Analogous maximum principle applies to varieties with bounded mean curvature.

Results extend classical maximum principles to higher codimension cases.

Abstract

We prove that an m-dimensional minimal variety in a Riemannian manifold cannot touch the boundary at a point where the sum of the smallest m principal curvatures is greater than 0. We also prove an analogous result for varieties with bounded mean curvature.