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Minimal networks on S^2

Xuyan Liu

Abstract

The minimal network problem is a classical topic in geometric measure theory and the calculus of variations, which aims to find networks of minimal length connecting given points. Most classical results are established in the Euclidean plane, while a complete theory for constant-curvature Riemannian manifolds remains to be developed. In this paper, we locally extend the theory of minimal networks and the calibration method from the Euclidean plane to the standard unit sphere \(S^2\). We redefine \(\mathbb{R}^2\)-valued co-vectors, differential forms, currents, and calibrations adapted to spherical geometry. Using exponential maps and local metric perturbation estimates, we prove that spherical minimal networks composed of great-circle arcs with \(120^\circ\) triple junctions are \textbf{locally length-minimizing only within sufficiently small geodesic balls} on \(S^2\), without obtaining global minimality results. Our work partially enriches the theory of minimal networks on constant-curvature spaces, and provides a theoretical reference and technical basis for future research on extending such results to higher-dimensional Riemannian manifolds and more general surfaces.

Minimal networks on S^2

Abstract

The minimal network problem is a classical topic in geometric measure theory and the calculus of variations, which aims to find networks of minimal length connecting given points. Most classical results are established in the Euclidean plane, while a complete theory for constant-curvature Riemannian manifolds remains to be developed. In this paper, we locally extend the theory of minimal networks and the calibration method from the Euclidean plane to the standard unit sphere . We redefine -valued co-vectors, differential forms, currents, and calibrations adapted to spherical geometry. Using exponential maps and local metric perturbation estimates, we prove that spherical minimal networks composed of great-circle arcs with triple junctions are \textbf{locally length-minimizing only within sufficiently small geodesic balls} on , without obtaining global minimality results. Our work partially enriches the theory of minimal networks on constant-curvature spaces, and provides a theoretical reference and technical basis for future research on extending such results to higher-dimensional Riemannian manifolds and more general surfaces.

Paper Structure

This paper contains 8 sections, 3 theorems, 39 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic concepts
  4. Properties
  5. Introduction to Minimal networks on $S^2$
  6. Main Results
  7. The Spherical Cap Version
  8. Acknowledgments

Key Result

Theorem 2.1

Let $\Gamma_*: G \to \mathbb{R}^2$ be a minimal network on a set of endpoints $\{p_1, \dots, p_n\} \subset \mathbb{R}^2$. Then there exists a 1-rectifiable current $\hat{T}$ with coefficients in $\mathcal{G}$ satisfying: $\blacktriangleleft$$\blacktriangleleft$

Figures (3)

  • Figure 1: Sketch of an immersed triple-junction network
  • Figure 2: Sketch of the standard unit sphere $S^2$
  • Figure 3: Schematic illustration of the local length-minimality of a spherical minimal network

Theorems & Definitions (18)

  • Definition 2.1: r7
  • Definition 2.2: r7
  • Definition 2.3: r7
  • Remark 2.1
  • Definition 2.4: r7
  • Theorem 2.1: r7
  • proof
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 8 more