Minimal Networks on Balls and Spheres for Almost Standard Metrics

Abstract

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