Stable geodesic nets in convex hypersurfaces

Abstract

We construct convex bodies that can be "captured by nets." More precisely, for each dimension $n\ge 2$, we construct a family of Riemannian $n$-spheres, each with a stable geodesic net, which is a stable 1-dimensional integral varifold. Small perturbations of a stable geodesic net must lengthen it. These stable geodesic nets are composed of multiple geodesic loops based at the same point, and also do not contain any closed geodesic. All of these Riemannian $n$-spheres are isometric to convex hypersurfaces of ${\mathbb{R}}^{n+1}$ with positive sectional curvature.

arXiv comment21 pages, 5 figures. To be published in the Journal of Geometric Analysis. Minor amendments were made, such as adding more details to the proofs of Lemma 4.2 and Proposition 4.6