On Berger's Isoperimetric Problem

2025/06/06 by Fan Kang, Kang, Fan · 1 citation
Mathematics · #Analytic and geometric function theory #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Point processes and geometric inequalities #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2506.05775

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

Abstract

Berger's isoperimetric problem asks if the flat equilateral torus is λ₁-maximal. In 1996, Nadirashvili first gave a positive answer. In this paper, we use El Soufi-Ilias-Ros's method and Bryant's result (arXiv:1507.01485) to give a new proof.

Citations

• On branched minimal immersions of surfaces by first eigenfunctions
• Metrics on a closed surface of genus two which maximize the first eigenvalue of the Laplacian
• On the conformal volume of 2-tori
• How large can the first eigenvalue be on a surface of genus two?
• Extremal metric for the first eigenvalue on a Klein bottle
• Extremal Metric for the First Eigenvalue on a Klein Bottle
• A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces

Cited by

• An improved upper bound for the second eigenvalue on tori

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