Minimal networks on balls and spheres for almost standard metrics

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.