Stable geodesic nets in convex hypersurfaces

TL;DR

This paper constructs convex hypersurfaces in higher dimensions that contain stable geodesic nets, which are resistant to small perturbations and do not include closed geodesics, advancing understanding of geodesic stability in convex geometry.

Contribution

The paper introduces the first examples of convex hypersurfaces with stable geodesic nets composed of multiple loops, expanding the class of known stable geodesic configurations.

Findings

Existence of stable geodesic nets in convex hypersurfaces for all dimensions n ≥ 2

Stable geodesic nets are composed of multiple loops based at the same point

These nets do not contain any closed geodesic

Abstract

We construct convex bodies that can be "captured by nets." More precisely, for each dimension $n \geq 2$, 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.