Minimal Networks on Balls and Spheres for Almost Standard Metrics

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