Existence of harmonic maps and eigenvalue optimization in higher dimensions

2022/07/27 by Mikhail Karpukhin, Daniel Stern, Karpukhin, Mikhail +1 · 3 citations
Computer Science · Engineering · #Advanced Mathematical Modeling in Engineering #Advanced Numerical Methods in Computational Mathematics #Differential Geometry (math.DG) #FOS: Mathematics #Spectral Theory (math.SP) #Topology Optimization in Engineering

paper · pdf · doi:10.48550/arxiv.2207.13635

openalex publication_date 2022/07/27 · openalex created_date 2022/07/30 · openalex updated_date 2026/07/28

Abstract

We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold (Mⁿ,g) of dimension n>2 to any closed, non-aspherical manifold N containing no stable minimal two-spheres. In particular, this gives the first general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets. In the special case of the round spheres N=\mathbbS^k, k≥ 3, we obtain a distinguished family of nonconstant harmonic maps M→ \mathbbS^k of index at most k+1, with singular set of codimension at least 7 for k sufficiently large. Furthermore, if 3≤ n≤ 5, we show that these smooth harmonic maps stabilize as k becomes large, and correspond to the solutions of an eigenvalue optimization problem on M, generalizing the conformal maximization of the first Laplace eigenvalue on surfaces.

Cited by

• Optimal eigenvalues on a metric graph with densities
• Eigenvalue optimization via a first-variation formula
• Rigidity in the Ginzburg--Landau approximation of harmonic spheres

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• Maximizing higher eigenvalues in dimensions three and above
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• Laplace and Steklov extremal metrics via n-harmonic maps