Minimal networks on S^2

TL;DR

This paper extends the theory of minimal networks from the Euclidean plane to the sphere, showing local length-minimizing properties of spherical networks with triple junctions.

Contribution

It adapts calibration methods and geometric tools to spherical geometry, establishing local minimality of great-circle arc networks on the sphere.

Findings

Spherical minimal networks with 120° triple junctions are locally length-minimizing.

The theory is extended using exponential maps and local metric estimates.

Global minimality on the sphere remains unproven.

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.