Partial result of Yau's Conjecture of the first eigenvalue in unit sphere $\mathbb{S}^{n+1}(1)$

TL;DR

This paper provides partial progress on Yau's Conjecture regarding the first eigenvalue of minimal hypersurfaces in the unit sphere, establishing new conditions and partial solutions to a longstanding geometric problem.

Contribution

It offers a partial proof of Yau's Conjecture for embedded minimal hypersurfaces in spheres and introduces a meaningful integral condition related to the eigenvalue problem.

Findings

Partial solution to Yau's Conjecture for minimal hypersurfaces

Establishment of a natural integral condition for eigenvalues

Corollary 1.3 confirms the significance of the integral condition

Abstract

In this paper, we partially solve Yau' Conjecture of the first eigenvalue of an embedded compact minimal hypersurface of unit sphere $\mathbb{S}^{n+1}(1)$, i.e., Corollary 1.2. In particular, Corollary 1.3 proves that the condition $\int_{\Omega_{1}}|\nabla u|^{2}=(n+1)\int_{\Omega_{1}}u^{2}$ is naturally true and meaningful in Corollary 1.2.