Minimal disks and compact hypersurfaces in Euclidean space

Abstract:

Let ${M^n}$ be a smooth connected compact hypersurface in $(n + 1)$-dimensional Euclidean space ${E^{n + 1}}$, let ${A^{n + 1}}$ be the unbounded component of ${E^{n + 1}} - {M^n}$, and let ${\kappa _1} \leqslant {\kappa _2} \leqslant \cdots \leqslant {\kappa _n}$ be the principal curvatures of ${M^n}$ with respect to the unit normal pointing into ${A^{n + 1}}$. It is proven that if ${\kappa _2} + \cdots + {\kappa _n} < 0$, then ${A^{n + 1}}$ is simply connected.