Existence of harmonic maps and eigenvalue optimization in higher dimensions

• 成果类型: Article
• 署名作者: Karpukhin, Mikhail; Stern, Daniel
• 署名单位: California Institute of Technology; University of Chicago
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-024-01247-3
• 发表日期: 2024
• 页码: 713-778
• 关键词: MINIMAL IMMERSIONS; PARTIAL REGULARITY; LAPLACIAN EIGENVALUE; SINGULAR SET; MIN-MAX; MANIFOLDS; METRICS

摘要

We prove the existence of nonconstant harmonic maps of optimal regularity from an arbitrary closed manifold (M-n,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=S-k, k >= 3, we obtain a distinguished family of nonconstant harmonic maps M -> S-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.