Minimal disks and compact hypersurfaces in Euclidean space

Abstract

Let M n {M^n} be a smooth connected compact hypersurface in ( n + 1 ) (n + 1) -dimensional Euclidean space E n + 1 {E^{n + 1}} , let A n + 1 {A^{n + 1}} be the unbounded component of E n + 1 − M n {E^{n + 1}} - {M^n} , and let κ 1 ⩽ κ 2 ⩽ ⋯ ⩽ κ n {\kappa _1} \leqslant {\kappa _2} \leqslant \cdots \leqslant {\kappa _n} be the principal curvatures of M n {M^n} with respect to the unit normal pointing into A n + 1 {A^{n + 1}} . It is proven that if κ 2 + ⋯ + κ n > 0 {\kappa _2} + \cdots + {\kappa _n} > 0 , then A n + 1 {A^{n + 1}} is simply connected.