Density of a minimal submanifold and total curvature of its boundary

Abstract

Given a piecewise smooth submanifold $Γ^{n-1} \subset \R^m$ and $p \in \R^m$, we define the {\em vision angle} $Π_p(Γ)$ to be the $(n-1)$-dimensional volume of the radial projection of $Γ$ to the unit sphere centered at $p$. If $p$ is a point on a stationary $n$-rectifiable set $Σ\subset \R^m$ with boundary $Γ$, then we show the density of $Σ$ at $p$ is $\leq$ the density at its vertex $p$ of the cone over $Γ$. It follows that if $Π_p(Γ)$ is less than twice the volume of $S^{n-1}$, for all $p \in Γ$, then $Σ$ 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 $Γ_i$ of $R^n_i, i=1,2$, a nonflat minimal submanifold spanned by $Γ:=Γ_1\cupΓ_2$ is embedded.