On spectral geometry of minimal parallel submanifolds

Abstract

We consider the problem of characterizing some minimal submanifolds using the spectrum0Spec of the Laplace-Beltrami operator acting on fucntions. In particular we characterize then-dimensional compact minimal totally real parallel submanifolds immersed in the complex projective spaceCPn, 3≤n≤6, by their0Spec in the class of all compact totally real minimal submanifolds ofCPn. Moreover, we characterize the Clifford torus\(S^1 \left( {\sqrt {\frac{1}{n}} } \right){\text{x}}S^{n - 1} \left( {\sqrt {\frac{{n - 1}}{n}} } \right)\) by its0Spec in the class of all compact minimal submanifolds of the Euclidean sphereSn+1(1).