Geodesic nets via eigenvalue optimisation

Duc Hoang Cao

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arXiv · 2508.16331

Geodesic nets via eigenvalue optimisation

Duc Hoang Cao

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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.

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Duc Hoang Cao. 2025-08-22. Geodesic nets via eigenvalue optimisation. https://arxiv.org/abs/2508.16331

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