Density of a minimal submanifold and total curvature of its boundary

TL;DR

This paper establishes a relationship between the density of minimal submanifolds at boundary points and the total curvature of their boundary, providing conditions under which such submanifolds are guaranteed to be embedded.

Contribution

It introduces the concept of vision angle and links boundary total curvature to the embeddedness of minimal submanifolds, extending classical results in geometric measure theory.

Findings

Density at boundary points is bounded by cone density over boundary

If vision angle is less than twice the sphere volume, the submanifold is embedded

Minimal submanifolds spanned by convex hypersurfaces in two planes are embedded

Abstract

Given a piecewise smooth submanifold $\Gamma^{n-1} \subset \R^m$ and $p \in \R^m$, we define the {\em vision angle} $\Pi_p(\Gamma)$ to be the $(n-1)$-dimensional volume of the radial projection of $\Gamma$ to the unit sphere centered at $p$. If $p$ is a point on a stationary $n$-rectifiable set $\Sigma \subset \R^m$ with boundary $\Gamma$, then we show the density of $\Sigma$ at $p$ is $\leq$ the density at its vertex $p$ of the cone over $\Gamma$. It follows that if $\Pi_p(\Gamma)$ is less than twice the volume of $S^{n-1}$, for all $p \in \Gamma$, then $\Sigma$ is an embedded submanifold. As a consequence, we prove that given two $n$-planes $R^n_1, R^n_2$ in $\R^m$ and two compact convex hypersurfaces $\Gamma_i$ of $R^n_i, i=1,2$, a nonflat minimal submanifold spanned by $\Gamma:=\Gamma_1\cup\Gamma_2$ is embedded.