Minimal networks on balls and spheres for almost standard metrics

Abstract

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

20 pages, 2 figures