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This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.This paper is aimed to summary some previous conclusions about the upper bound for the eigenvalues of the Laplace-Beltrami operator on sphere and to give more details about their proof.In addition, we propose some of our own thinking about some open problem about the eigenvalues of Laplace operator.