Minimal Networks on Balls and Spheres for Almost Standard Metrics

Abstract We study the existence of minimal networks in the unit sphere $${\textbf{S}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> and the unit ball $${\textbf{B}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> of $${\textbf{R}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of $$\theta $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>θ</mml:mi> </mml:math> -networks in $${\textbf{S}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>S</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> and triods in $${\textbf{B}}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>B</mml:mi> </mml:mrow> <mml:mi>d</mml:mi> </mml:msup> </mml:math> , jointly with the Lusternik–Schnirelmann category.