Density of a minimal submanifold and total curvature of its boundary

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.