First-eigenvalue maximization and inflation of maps

2025/06/06 by Shin Nayatani, Nayatani, Shin
Mathematics · #53C42 #58J50 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Mathematical Dynamics and Fractals #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2506.05681

openalex publication_date 2025/06/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Given a compact manifold equipped with a volume element and a Riemannian metric, we formulate and study a dual pair of optimization problems: one concerning smooth maps from the manifold into the Hilbert space l² and the other concerning the smallest positive eigenvalue of the Bakry-Emery Laplacian. We present examples of manifolds for which these problems can be solved explicitly. We also prove a Nadirashvili-type theorem.

Citations

• The first eigenvalue of the Laplacian on orientable surfaces
• Sharp asymptotics of the first eigenvalue on some degenerating surfaces
• Isoperimetric rigidity and distributions of 1-Lipschitz functions
• Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian
• Existence and regularity of maximal metrics for the first Laplace eigenvalue on surfaces
• Isoparametric foliation and Yau conjecture on the first eigenvalue, II
• How large can the first eigenvalue be on a surface of genus two?
• First eigenvalue of symmetric minimal surfaces in \mathbbS³

Related

• Eigenvalue estimates and applications on weighted manifolds
• Maximizing one Laplace eigenvalue on n-dimensional manifolds
• Maximization of the first nontrivial eigenvalue on the surface of genus two
• An alternative proof of lower bounds for the first eigenvalue on manifolds
• Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds