Convergence of Ricci Flow on a Class of Warped Product Metrics
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Convergence of Ricci Flow on a Class of Warped Product Metrics Tobias Marxen1 Received: 8 April 2019 © Mathematica Josephina, Inc. 2019
Abstract We consider Ricci flow starting from warped product manifolds R × N , k0 + g02 g N , whose typical fibre (N , g N ) is closed and Ricci flat. Here k0 is a Riemannian metric on R and g0 : R → R is positive. Under a mild condition, we show that (i) if the initial metric is asymptotic to the Ricci flat metric k0 + c2 g N , where c > 0, the solution of the Ricci flow converges smoothly uniformly to a Ricci flat metric as t → ∞, up to pullback by a family of diffeomorphisms, and (ii) if the initial manifold is asymptotic to the real line, then the solution converges uniformly (in Gromov Hausdorff distance) to the real line as t → ∞. In the course of the proof, we establish an averaging and a convergence result for the heat equation on noncompact manifolds with timedependent metric, that might be of independent interest. Keywords Ricci flow · Warped product · Noncompact Mathematics Subject Classification 53C44 (Primary) · 58D19 (Secondary)
1 Introduction Given a manifold M, a family of Riemannian metrics h(t), t ∈ [0, T ) on M, where 0 < T ≤ ∞, evolves by Ricci flow, starting from an initial metric h 0 , if
d dt h(t)( p) = h(0) = h 0 .
−2 Rich(t) ( p) for all p ∈ M, t ∈ [0, T ),
(1.1)
Supported by the DFG (German Research Foundation) Priority Programme “Geometry at Infinity”.
B 1
Tobias Marxen [email protected] Universität Oldenburg, Oldenburg, Germany
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T. Marxen
Here Rich(t) denotes the Ricci tensor with respect to h(t). The Ricci flow, introduced by Hamilton [14] in 1982, is a nonlinear geometric partial differential equation. Thus, a natural approach to study the flow is to consider solutions with symmetries. Hamilton [16, Sect. 11] and Carfora et al. [4] each investigated Ricci flow on a class of metrics invariant under a free, isometric T 2 action on the three-torus T 3 , and proved longtime existence and convergence to a flat metric. Hamilton and Isenberg [17] considered a class of metrics on twisted T 2 bundles over S 1 corresponding to the class in [4] and showed quasi-convergence (for a definition, see Knopf [21]) of the flow. Generalizing the results above, Lott and Sesum [23] described the Ricci flow on (twisted) T 2 bundles over S 1 and also showed convergence on warped products with S 1 fibres over a compact surface. For more results about the Ricci flow on manifolds with symmetries see below and the introduction in [23]. In each case above the Ricci flow is investigated on a class of compact three-manifolds with symmetries. This raises the following Question Under which conditions does the Ricci flow (quasi-)converge in the corresponding higher dimensional and/or noncompact cases? In this paper, we give a partial answer to this question. We consider a class of manifolds which are noncompact, n + 1-dimensional analogues of Hamilton’s. More precisely, we let M = R× N , equipped with a warped product metric h 0 = k0 +g02 g N , where k
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