The characterization of eigenfunctions for Laplacian operators
نویسنده
چکیده
In this paper, we consider the characterization of eigenfunctions for Laplacian operators on some Riemannian manifolds. Firstly we prove that for the space form (M K , gK) with the constant sectional curvature K, the first eigenvalue of Laplacian operator λ1 (M K) is greater than the limit of the first Dirichlet eigenvalue of Laplacian operator λ1 (BK (p, r)). Based on this, we then present a characterization of the Ricci soliton being an n-dim space form by the eigenfunctions corresponding to the first eigenvalue of Laplacian operator, which gives a generalization of an interesting result by Cheng in [3] from 2-dim to n-dim. Moreover, this result also gives a partly proof of a conjecture by Hamilton that a compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. M.S.C. 2010: 58G25, 35P05.
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