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In this paper, we deal with comparison theorems for the first eigenvalue of the Schrödinger 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 μ1 (B(p, r)) of a given Schrödinger operator. We then prove that, for the space form Mn K with constant sectional curvature K, the first eigenvalue of the Laplacian operator λ1(M K) is greater than the limit of the corresponding first Dirichlet eigenvalue λ1 (BK(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 (M3, g) satisfies the Ricci pinching condition Rc ≥ εRg, where R > 0 and ε is a positive constant, then it is compact. In fact, our result is also true for general n-dim manifolds.

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