Characterizations of the compactness of Riemannian manifolds by eigenfunctions, and a partial proof of a conjecture by Hamilton

In this paper, we deal with comparison theorems for the first eigenvalue of the Schrodinger operator, and we present some criteria for the compactness of a Riemannian manifold in terms of the eigenfunctions of its Laplacian. Firstly, we establish a comparison theorem for the first Dirichlet eigenvalue mu(D)(1)(B (p, r)) of a given Schrodinger operator. We then prove that, for the space form M-K(n) with constant sectional curvature K, the first eigenvalue of the Laplacian operator lambda(1) (M-K(n)) is greater than the limit of the corresponding first Dirichlet eigenvalue mu(D)(1)(B-K (p, r)). Based on these, we present a characterization of a compact gradient shrinking Ricci soliton locally being an n-dim space form by the first eigenfunctions of the Laplacian operator, which gives a generalization of an interesting result by Cheng [4] from 2-dim to n-dim. This result also gives a partial proof of a conjecture by Hamilton [7] that a compact gradient shrinking Ricci soliton with positive curvature operator must be Einstein. Finally, we derive a criterion of the compactness of manifolds, which gives a partial proof of another conjecture by Hamilton [6] that, if a complete Riemannian 3-manifold (M-3, g) satisfies the Ricci pinching condition R-c >= epsilon Rg, where R > 0 and epsilon is a positive constant, then it is compact. In fact, our result is also true for general n-dim manifolds.