Density of a minimal submanifold and total curvature of its boundary

Abstract

Given a piecewise smooth submanifold ${\Gamma }^{n-1}\subset {\mathbb{R}}^m$ and $p\in {\mathbb{R}}^m$, 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 $\Sigma \subset {\mathbb{R}}^m$ with boundary $\Gamma$, then we show the density of $\Sigma$ at $p$ is $\le$ 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 $R_1^n,R_2^n$ in ${\mathbb{R}}^m$ and two compact convex hypersurfaces ${\Gamma }_i$ of $R_i^n,i=1,2$, a nonflat minimal submanifold spanned by $\Gamma :={\Gamma }_1\cup {\Gamma }_2$ is embedded.