Density of a minimal submanifold and total curvature of its boundary

Given a piecewise smooth submanifold $\Gamma^{n-1} \subset \mathbb{R}^m$ and $p \in \mathbb{R}^m$, we define the 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 $\leq$ 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 $\mathbb{R}^n_1 , \mathbb{R}^n_2$ in $\mathbb{R}^m$ and two compact convex hypersurfaces $\Gamma_i$ of $\mathbb{R}^n_i , i = 1, 2$, a nonflat minimal submanifold spanned by $\Gamma := \Gamma_1 \cup \Gamma_2$ is embedded.

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The first-named author was supported in part by KRF-2007-313-C00057.

Received 12 January 2012

Published 13 September 2017