Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Minimal networks on balls and spheres for almost standard metrics

Luciano Sciaraffia

Abstract

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 $θ$-networks in $\mathbf{S}^d$ and triods in $\mathbf{B}^d$, jointly with the Lusternik--Schnirelmann category.

Minimal networks on balls and spheres for almost standard metrics

Abstract

We study the existence of minimal networks in the unit sphere and the unit ball of endowed with Riemannian metrics close to the standard ones. We employ a finite-dimensional reduction method, modelled on the configuration of -networks in and triods in , jointly with the Lusternik--Schnirelmann category.
Paper Structure (5 sections, 16 theorems, 89 equations, 2 figures)

This paper contains 5 sections, 16 theorems, 89 equations, 2 figures.

Table of Contents

  1. Introduction and main results
  2. Definitions and setup
  3. A reduction method and proof of Theorem \ref{['thm:mainB']}
  4. Proof of Theorem \ref{['thm:mainS']}
  5. Appendix

Key Result

Theorem 1.1

Let $d \geq 2$ be an integer, and let $g_0$ be the standard round metric on the sphere $\mathbf{S}^d$. Then for any Riemannian metric $g$ on $\mathbf{S}^d$ sufficiently close to $g_0$ in the $C^2$ topology there exist at least four minimal $\theta$-networks on $(\mathbf{S}^d,g)$, when $d \neq 3,4$,

Figures (2)

  • Figure 1: The three possible minimal networks which divide a sphere into three regions: the $\theta$-network, the eyeglasses, and the figure 8.
  • Figure 2: The three possible minimal networks in a strictly convex domain: the triod, the double triod, and the hexagonal cell.

Theorems & Definitions (35)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Proposition 2.1
  • Definition 2.2: Minimal network
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 25 more