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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 Riemannianmetrics close to the standard ones. We employ a finite-dimensional reductionmethod, modelled on the configuration of $\theta$-networks in $\mathbf{S}^d$and triods in $\mathbf{B}^d$, jointly with the Lusternik--Schnirelmanncategory.

Minimal networks on balls and spheres for almost standard metrics