Value of first eigenvalue of some minimal hypersurfaces embedded in the unit sphere

Abstract

We prove that the first nonzero eigenvalue of the Laplace-Beltrami operator of equator-like minimal submanifold embedded in the sphere $ S^{n+1} $ is equal to $ n $. The proof uses the spectral properties of the heat kernel operator corresponding to the submanifold.

Keywords

• minimal submanifolds
• laplace operator
• eigenvalues
• heat kernel