Mathematics – Differential Geometry
Scientific paper
2010-04-19
Mathematics
Differential Geometry
11 pages, We changed the title and added a section that exhibits the relation between our example and the question posed by Br
Scientific paper
10.1007/s00526-011-0466-z
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 with the following properties: (1) B has strictly convex boundary. (2) There exists a complete nonconstant geodesic. (3) There does not exist a closed geodesic in B.
Bangert Victor
Roettgen Nena
No associations
LandOfFree
Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Isoperimetric Inequalities for Minimal Submanifolds in Riemannian Manifolds: A Counterexample in Higher Codimension will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-59793