On spectral characterizations of minimal hypersurfaces in a sphere

Abstract

Let M be a closed minimal hypersurface in an Euclidean sphere Sⁿ⁺¹(1). We first prove that a minimal isoparametric hypersurface M in a 4-dimensional sphere is completely determined by its spectrum Spec^p(M), here p∈{0, 1, 2, 3}. In higher dimensional sphere, we prove that if Spec^p(M)=Spec^p(M_{m, n−m}) for p=0, 1, where
M_{m, n−m}=S^m(√{\frac{m}{n}})×S^n−m(√{\frac{n−m}{n}})
is a Clifford torus, then M is M_{m, n−m}. Furthermore, we prove that M_{n, n}→S²ⁿ⁺¹(1) (n{≥}4) is also characterized by Spec^p(M_{n, n}) for some p=p(n).