Maximum principles for submanifolds of arbitrary codimension and bounded mean curvature

Abstract

We construct quadratic forms on \(\mathbb{R}^{n + k}\) which are subharmonic on any n-dimensional minimal submanifold in \(\mathbb{R}^{n + k}\) and, more generally, on submanifolds of bounded mean curvature. This leads to nonexistence results for connected n-dimensional minimal submanifolds in \(\mathbb{R}^{n + k}\) as well as to necessary conditions for the existence of connected submanifolds of bounded mean curvature with arbitrary codimension. Furthermore we discuss a barrier principle for n-dimensional submanifolds in \(\mathbb{R}^{n + k}\) of bounded mean curvature.