Minimal Networks on Balls and Spheres for Almost Standard Metrics

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