Regularity estimates on harmonic eigenmaps with arbitrary number of coordinates

2025/08/14 by Petrides, Romain
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.2508.10448

Abstract

We revisit the well-established regularity estimates on harmonic maps on surfaces to question their independence with respect to the dimension of the target manifold. We are mainly interested in harmonic maps into target ellipsoids, that we call Laplace harmonic eigenmaps. These maps are related to critical metrics in the context of eigenvalue optimization. The tools that we gather here are useful to handle convergence of almost critical metrics via Palais-Smale sequences of (almost harmonic) eigenmaps. They could also be a preliminary step for a general regularity theory for critical points of infinite combinations of eigenvalues.

Citations

• Critical metrics of eigenvalue functionals via Clarke subdifferential
• Conservation Laws for p-Harmonic Systems with Antisymmetric Potentials and Applications
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• Regularity and quantification for harmonic maps with free boundary
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Related

• Existence of harmonic maps and eigenvalue optimization in higher dimensions
• Laplace and Steklov extremal metrics via n-harmonic maps
• Convergence, optimization and stability of singular eigenmaps
• A survey of classical results on harmonic maps
• The regularity of harmonic maps into spheres and applications to Bernstein problems