TL;DR
This paper provides a new proof that the flat equilateral torus is maximally optimized for the first Laplacian eigenvalue, using advanced geometric methods and recent results.
Contribution
It offers an alternative proof of Berger's isoperimetric problem leveraging modern techniques and previous findings.
Findings
Confirmed the flat equilateral torus is λ₁-maximal.
Introduced a new proof method based on El Soufi-Ilias-Ros's approach.
Utilized Bryant's recent result to support the proof.
Abstract
Berger's isoperimetric problem asks if the flat equilateral torus is -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.
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Taxonomy
TopicsAnalytic and geometric function theory · Point processes and geometric inequalities · Geometric Analysis and Curvature Flows