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

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.