Geodesic nets via eigenvalue optimisation

TL;DR

This paper investigates the relationship between geodesic nets and quantum graphs through spectral optimization, revealing how critical metrics relate to minimal immersions and identifying obstructions to their existence.

Contribution

It introduces a novel connection between spectral theory, quantum graphs, and geodesic nets, providing new insights into critical metrics and their geometric properties.

Findings

Critical metrics induce isometric minimal immersions to spheres.

Geodesic nets can be derived from optimal quantum graphs.

Obstructions to the existence of certain critical metrics are identified.

Abstract

We explore a connection between geodesic nets and quantum graphs optimising certain functionals from spectral theory. For surfaces, critical metrics for the normalised $k^{\mathrm{th}}$ eigenvalue of the Laplacian give rise to isometric minimal immersions to a unit sphere. In this spirit we obtain geodesic nets from optimal quantum graphs, and obstructions to the existence of critical metrics.