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

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.