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

Abstract

In this paper, we deal with comparison theorems for the first eigenvalue of the Schr\"{o}dinger 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 $\mu _1^D (B (p,r))$ of a given Schr\"{o}dinger 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 $\lambda _1^D(B_K(p,r))$. 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 \cite {4} from 2-dim to $n$-dim. This result also gives a partial proof of a conjecture by Hamilton \cite {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 \cite {6} that, if a complete Riemannian 3-manifold $(M^3, g)$ satisfies the Ricci pinching condition $Rc \ge \varepsilon Rg$, where $R > 0$ and $\varepsilon $ is a positive constant, then it is compact. In fact, our result is also true for general $n$-dim manifolds.