Isoperimetric inequalities for minimal submanifolds in Riemannian manifolds: a counterexample in higher codimension |
| |
Authors: | Victor Bangert Nena R?ttgen |
| |
Institution: | 1. Mathematisches Institut, Abteilung für Reine Mathematik, Albert-Ludwigs-Universit?t, Eckerstr. 1, 79104, Freiburg im Breisgau, Germany
|
| |
Abstract: | For compact Riemannian manifolds with convex boundary, B. White proved the following alternative: either there is an isoperimetric inequality for minimal hypersurfaces or there exists a closed minimal hypersurface, possibly with a small singular set. There is the natural question if a similar result is true for submanifolds of higher codimension. Specifically, B. White asked if the non-existence of an isoperimetric inequality for k-varifolds implies the existence of a nonzero, stationary, integral k-varifold. We present examples showing that this is not true in codimension greater than two. The key step is the construction of a Riemannian metric on the closed four–dimensional ball B 4 with the following properties: (i) B 4 has strictly convex boundary. (ii) There exists a complete nonconstant geodesic ${c : \mathbb{R} \to B^4}$ . (iii) There does not exist a closed geodesic in B 4. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|