Geodesic nets via eigenvalue optimisation

2025/08/22 by Cao, Duc Hoang · 1 citation
#34L05 #34L40 #58E11 #81Q35 #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2508.16331

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^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.

Citations

• Laplace and Steklov extremal metrics via n-harmonic maps
• Minimal surfaces and eigenvalue problems
• Laplacian eigenvalues functionals and metric deformations on compact manifolds

Cited by

• Optimal eigenvalues on a metric graph with densities

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