Asymptotically cylindrical Ricci-flat manifolds

Sema Salur
2006 Proceedings of the American Mathematical Society  
Asymptotically cylindrical Ricci-flat manifolds play a key role in constructing Topological Quantum Field Theories. It is particularly important to understand their behavior at the cylindrical ends and the natural restrictions on the geometry. In this paper we show that an orientable, connected, asymptotically cylindrical manifold (M, g) with Ricci-flat metric g can have at most two cylindrical ends. In the case where there are two such cylindrical ends, then there is reduction in the holonomy
more » ... roup Hol(g) and (M, g) is a cylinder. 3049 License or copyright restrictions may apply to redistribution; see 3050 SEMA SALUR l ≤ 2, and in the case when l = 2 there is reduction in the holonomy group Hol(g) and (M, g) is a cylinder. Remark 1.2. The proof of this theorem also follows from the Cheeger-Gromoll splitting theorem [2] . The key ingredients of the splitting theorem are the maximum principle for continuous functions, Laplacian comparison theorems, constructions of rays, lines and Busemann functions. For more details on the subject see [11] . In this paper we give an alternative proof and show that the reduction in holonomy can be obtained by just using the analytic set-up for Fredholm properties of an elliptic operator on noncompact manifolds. This analytic set-up was developed by Lockhart and McOwen, [8] and by Melrose, [9], [10]. Asymptotically cylindrical manifolds In this section we first introduce the definitions of cylindrical and asymptotically cylindrical Riemannian manifolds. Definition 2.1. An n-dimensional Riemannian manifold (M 0 , g 0 ) is called cylindrical if M 0 = X × R and g 0 is compatible with this product structure, that is, where X is a compact, connected (n − 1)-manifold with Riemannian metric g X . Definition 2.2. A connected, complete n-manifold (M, g) with l cylindrical ends is called asymptotically cylindrical with decay rate β = (β 1 , . . . , β l ) ∈ R l , β j < 0 for 1 ≤ j ≤ l, if there exist cylindrical n-manifolds (M 0i , g 0i ) with M 0i = X i × R for connected X i , a compact subset K ⊂ M , a real number R, and diffeomorphisms Ψ i : where ∇ 0i is the Levi-Civita connection of the cylindrical metric g 0i on M 0i = X i × R. Remark 2.3. For simplicity, we will assume that the decay rates are all equal for each cylindrical end, that is, β j = β k , 1 ≤ j, k ≤ l.
doi:10.1090/s0002-9939-06-08313-4 fatcat:qyhvmqrb4jbnjlquojmredbxsq