Work

A lower bound for the first eigenvalue of a minimal hypersurface in the sphere

Year: 2025

Type: article

Abstract: Let \Sigma be a closed embedded minimal hypersurface in the unit sphere \mathbb{S}^{m+1} and let \Lambda=\max_{\Sigma}|A| be the norm of its second fundamental form. In this work, we prove that the first eigenvalue of the Laplacian of \Sigma satisfies \lambda_{1}(\Sigma) > \frac{m}{2}+\frac{m(m+1)}{32 (12 \Lambda+m+11)^{2}+8}, and \lambda_{1}(\Sigma)=m when \Lambda\le\sqrt{m} . In particular, this estimate improves the one obtained recently in Duncan–Sire–Spruck (2024). The proof of our main result is based on a Rayleigh quotient estimate for a harmonic extension of an eigenfunction of the Laplacian of \Sigma in the spirit of Choi and Wang (1983).

Source: Revista Matemática Iberoamericana

DOI: 10.4171/rmi/1587

Authors: Asun Jiménez, Carlos Tapia Chinchay, Detang Zhou

Institution: Universidade Federal Fluminense

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: Coordenação de Aperfeiçoamento de Pessoal de Nível Superior, Conselho Nacional de Desenvolvimento Científico e Tecnológico, Fundação Carlos Chagas Filho de Amparo à Pesquisa do Estado do Rio de Janeiro

Awards: G1840770272, G2584274330, G5089450235, G6443130175