Minimal Networks on Balls and Spheres for Almost Standard Metrics

Abstract

We study the existence of minimal networks in the unit sphere \({\textbf{S}}^d\) and the unit ball \({\textbf{B}}^d\) of \({\textbf{R}}^d\) endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of \(\theta \)-networks in \({\textbf{S}}^d\) and triods in \({\textbf{B}}^d\), jointly with the Lusternik–Schnirelmann category.