Spectrum of the Laplacian on quaternionic Kähler manifolds

Abstract

Let M 4n be a complete quaternionic Kahler manifold with scalar curvature bounded below by −16n(n+ 2). We get a sharp estimate for the first eigenvalue λ1(M) of the Laplacian, which is λ1(M) ≤ (2n + 1) . If the equality holds, then either M has only one end, or M is diffeomorphic to R × N with N given by a compact manifold. Moreover, if M is of bounded curvature, M is covered by the quaterionic hyperbolic space QHn and N is a compact quotient of the generalized Heisenberg group. When λ1(M) ≥ 8(n+2) 3 , we also prove that M must have only one end with infinite volume. 0. Introduction LetM be a complete n-dimensional Riemannian manifold with Ricci curvature bounded below by −(n−1). It is well known from Cheng [Ch] that the first eigenvalue λ1(M) satisfies λ1(M) ≤ (n− 1)2 4 . In [LW3], Li and Wang proved an analogous theorem for complete Kahler manifolds. They showed that if M2n is a complete Kahler manifold of complex dimension n with holomorphic bisectional curvature BKM bounded below by −1, then the first eigenvalue λ1(M) satisfies λ1(M) ≤ n . Here BKM ≥ −1 means that Rīijj ≥ −(1 + δij) for any unitary frame e1, . . . , en. In this paper, we prove the corresponding Laplacian comparison theorem for a quaterionic Kahler manifold M4n. As an application we get The second author was partially supported by NSF grant #DMS-0503735. The third author was partially supported by CAPES and CNPq of Brazil. Received 03/27/2007. 295 296 S. KONG, P. LI & D. ZHOU the sharp estimate λ1(M) for a complete quaterionic Kahler manifold M4n with scalar curvature bounded below by −16n(n+ 2) as λ1(M) ≤ (2n+ 1) . It is an interesting question to ask what one can say about those manifolds when the above inequalities are realized as equalities. In works of Li and Wang [LW1] and [LW2], the authors obtained the following theorems. The first was a generalization of the theory of Witten-Yau [WY], Cai-Galloway [CG], and Wang [W] for conformally compact manifolds. The second was to answer the aforementioned question. Theorem 0.1. Let M be a complete Riemannian manifold of dimension n ≥ 3 with Ricci curvature bounded below by −(n − 1). If λ1(M) ≥ n− 2, then either (1) M has only one infinite volume end; or (2) M = R ×N with warped product metric of the form dsM = dt 2 + cosh t dsN , where N is an (n−1)-dimensional compact manifold of Ricci curvature bounded below by λ1(M). Theorem 0.2. Let M be a complete Riemannian manifold of dimension n ≥ 2 with Ricci curvature bounded below by −(n − 1). If λ1(M) ≥ (n−1) 4 , then either (1) M has no finite volume end; or (2) M = R ×N with warped product metric of the form dsM = dt 2 + e dsN , where N is an (n− 1)-dimensional compact manifold of nonnegative Ricci curvature. In [LW3] and [LW5], Li and Wang also consider the Kahler case. They proved the following theorems. Theorem 0.3. Let M be a complete Kahler manifold of complex dimension n ≥ 1 with Ricci curvature bounded below by RicM ≥ −2(n+ 1). If λ1(M) > n+1 2 , then M must have only one infinite volume end. Theorem 0.4. Let M be a complete Kahler manifold of complex dimension n ≥ 2 with holomorphic bisectional curvature bounded by BKM ≥ −1. If λ1(M) ≥ n 2, then either (1) M has only one end; or SPECTRUM ON QUATERNIONIC KAHLER MANIFOLDS 297 (2) M = R × N with N being a compact manifold. Moreover, the metric on M is of the form dsM = dt 2 + e ω 2 + e 2t 2n ∑ i=3 ω i , where {ω2, ω3, . . . , ω2n} are orthonormal coframe of N with Jdt = ω2. If M has bounded curvature, then we further conclude that M is covered by CH and N is a compact quotient of the Heisenberg group. In [LW5], the authors pointed out that the assumption on the lower bound of λ1(M) in Theorem 0.3 is sharp, since one can construct M of the form M = Σ×N satisfying RicM ≥ −2(n+ 1) (0.1) and λ1(M) = n+ 1 2 (0.2) withN being a compact Kahler manifold and Σ being a complete surface with at least two infinite volume ends. However, it is still an open question to characterize all those complete Kahler manifolds satisfying conditions (0.1) and (0.2). In Sections 4 and 5, we will prove the following quaternionic Kahler versions of the above theorems. Theorem 0.5. Let (M4n, g) be a complete quaternionic Kahler manifold with scalar curvature satisfying SM ≥ −16n(n+ 2). If λ1(M) ≥ 8(n+2) 3 , then M must have only one infinite volume end. Theorem 0.6. Let (M4n, g) be a complete quaternionic Kahler manifold with scalar curvature satisfying SM ≥ −16n(n+ 2). If λ1(M) ≥ (2n+ 1) 2, then either (1) M has only one end, or (2) M is diffeomorphic to R × N , where N is a compact manifold. Moreover, the metric is given by the form dsM = dt 2 + e 4 ∑