The strong elliptic maximum principle for vector bundles and applications to minimal maps

TL;DR

This paper establishes a strong elliptic maximum principle for vector bundle sections over Riemannian manifolds and applies it to derive rigidity and Bernstein type theorems for minimal maps in higher codimension.

Contribution

It introduces a new maximum principle for vector bundles and demonstrates its applications in proving geometric rigidity and minimal surface theorems.

Findings

Proved a strong elliptic maximum principle for vector bundle sections.

Derived rigidity theorems for minimal maps between Riemannian manifolds.

Established Bernstein type theorems in higher codimension.

Abstract

Based on works by Hopf, Weinberger, Hamilton and Evans, we state and prove the strong elliptic maximum principle for smooth sections in vector bundles over Riemannian manifolds and give some applications in Differential Geometry. Moreover, we use this maximum principle to obtain various rigidity theorems and Bernstein type theorems in higher codimension for minimal maps between Riemannian manifolds.