On the curvature estimates for the conformal Ricci flow
Introduction
A. Fischer introduced the conformal Ricci flow as a modified Ricci flow that preserves scalar curvature constant (see [5]). The definition of the conformal Ricci flow is as follows. Definition 1.1 Suppose that is a smooth n-dimensional Riemannian manifold with a constant scalar curvature . The conformal Ricci flow on M is defined as a family of metrics and functions such that with initial condition , where Ric is the Ricci curvature.
A. Fischer proved that the Yamabe constant monotonically increases under the conformal Ricci flow on the compact manifold of negative Yamabe type (see [5]). This suggests that the conformal Ricci flow may potentially be useful in finding Einstein metrics. It was also shown in [5] that the conformal Ricci flow exists for a short time on compact manifolds of negative Yamabe type. Using DeTurck's trick and implicit function theorem, Lu Peng, Qing Jie, and Zheng Yu generalized the existence results in [5] to include compact manifolds of positive Yamabe type as long as the operator is invertible (see [12]). Lu Peng, Qing Jie, and Zheng Yu proved the short-time existence on asymptotically hyperbolic manifolds and asymptotically flat manifolds (see [12], [13]). They also obtained the local Shi's type curvature derivative estimate for the conformal Ricci flow.
Since the conformal Ricci flow is similar to the Ricci flow, we expect that the Ricci flow results also hold for the conformal Ricci flow. The curvature estimates and extension problems for the Ricci flow have been studied for a long time. Hamilton proved that if the curvature operator of the Ricci flow solution is uniformly bounded on a finite interval , the solution can be extended to a larger interval (see [3]). Šešum generalized Hamilton's result and obtained that if the Ricci curvature of the solution to the Ricci flow is uniformly bounded on a finite interval , the curvature operator is also bounded (see [9]). But Šešum's result is based on blow-up analysis and can not be used to obtain the curvature operator's explicit bound by the Ricci curvature's bound. The extension problem has also been studied by many papers (see [1],[2],[4],[7],[8],[10],[11],[14],[15],[16],[17]). Recently, Kotschwar, Munteanu, and Wang Jiaping proved that the curvature tensor of the solution to the Ricci flow could be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensor, and the elapsed time T (see [6]). In this paper, we prove a similar result for the conformal Ricci flow. The following is our main result.
Theorem 1.2 Let be a smooth n-dimensional Riemannian manifold with constant scalar curvature . Let be a conformal Ricci flow with the initial condition . Suppose that there exist constants , and a point such that the ball is compactly contained in M and Then there are constants , depending only on the dimension n, such that where and is the Weyl tensor of the initial metric .
On the compact manifold, the curvature operator remains bounded so long as the Ricci curvature is bounded. Corollary 1.3 Suppose that is a smooth solution to the conformal Ricci flow on a compact manifold with , satisfying and then
Section snippets
The main estimates
The Kulkarni-Nomizu product is given by where are symmetric tensors. , , R, W denote the curvature operator, Ricci curvature, scalar curvature, and Weyl tensor, respectively. For the given tensor fields V and W, the notation represents some weighted sum of contractions with respect to the metric with coefficients bounded by universal constants. We first recall the equations for the curvature tensor along the conformal Ricci flow. Lemma 2.1
Proof of the main theorem
Suppose . Since and , , we have Consider the cut-off function which is Lipschitz with . From (38), it follows that Apply Proposition 2.2 to get By the Young inequality and the estimate (40), we have
Acknowledgments
This work is partially supported by National Natural Science Foundation of China (Grant number: 11971358). We are grateful to the anonymous reviewers for pointing out an error in the previous version of this paper.
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