The First Eigenvalue of Embedded Minimal Hypersurfaces in the Unit Sphere I: Yau's Conjecture

TL;DR

This paper proves Yau's conjecture that the first non-zero eigenvalue of the Laplace-Beltrami operator on embedded minimal hypersurfaces in the unit sphere equals the hypersurface's dimension, using variational methods.

Contribution

It provides a rigorous proof of Yau's conjecture for the first eigenvalue, resolving a long-standing open problem in differential geometry.

Findings

First eigenvalue equals the hypersurface dimension

Established several rigidity theorems

Confirmed Yau's conjecture in the embedded case

Abstract

In this paper, by meticulously constructing a minimizing sequence within a suitable Sobolev space and leveraging the variational principle, we establish that the first non-zero eigenvalue of the Laplace-Beltrami operator on an embedded minimal hypersurface in the unit sphere equals the dimension of the hypersurface. This result furnishes an affirmative resolution to a renowned conjecture posed by Yau, which had remained unresolved for an extended period. As some important applications, several rigidity theorems are established via eigenvalue characterization.