On the curvature estimates for the conformal Ricci flow

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Abstract

In this paper, we study the curvature estimates of the conformal Ricci flow on Riemannian manifolds. We show that the norm of the Weyl tensors of any smooth solution to the conformal Ricci flow can be explicitly estimated in terms of its initial values on a given ball, a local uniform bound on the Ricci tensors, and the potential function. On the compact manifold, the curvature operator remains bounded so long as the Ricci curvature is bounded.

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 (M,g0) is a smooth n-dimensional Riemannian manifold with a constant scalar curvature R0<0. The conformal Ricci flow on M is defined as a family of metrics g(t) and functions p(x,t) such that{gt+2(RicR0ng)=2p(x,t)g,(x,t)M×[0,T),R(g(t))=R0,(x,t)M×[0,T), with initial condition g(0)=g0, where Ric is the Ricci curvature.

It was shown in [5] that on compact manifolds, the conformal Ricci flow is equivalent to{gt+2(RicR0ng)=2pg,(x,t)M×[0,T),(n1)Δp+R0p=|RicR0ng|2,(x,t)M×[0,T).

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 (n1)Δg(t)+R0 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 [0,T), the solution can be extended to a larger interval [0,T+ϵ) (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 [0,T), 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 (M,g0) be a smooth n-dimensional Riemannian manifold with constant scalar curvature R0=1. Let (M,g(t),p(x,t)) be a conformal Ricci flow with the initial condition g(0)=g0. Suppose that there exist constants A,K>0, and a point x0M such that the ball Bg(0)(x0,A/K) is compactly contained in M and|Ric|K,|p|K2,onBg(0)(x0,AK)×[0,T). Then there are constants c,α,β>0, depending only on the dimension n, such that|Rm|(x0,T)cecK5(T+A)(1+((KT)1+A2)β+(Λ0K)α)(Λ0+K2(1+A2)), where Λ0:=supBg(0)|W|(x,0) and W(x,0) is the Weyl tensor of the initial metric g(0).

On the compact manifold, the curvature operator remains bounded so long as the Ricci curvature is bounded.

Corollary 1.3

Suppose that (M,g(t),p(x,t)),t[0,T) is a smooth solution to the conformal Ricci flow on a compact manifold with R0=1, satisfying|Ric|(x,t)K,(x,t)M×[0,T), andΛ:=supxM|Rm|(x,0)<, thensupM×[0,T)|Rm|(x,t)<.

Section snippets

The main estimates

The Kulkarni-Nomizu product is given by(AB)ijkl=AikBjl+AjlBikAilBjkAjkBil, where A,B are symmetric tensors. Rijkl, Rij, R, W denote the curvature operator, Ricci curvature, scalar curvature, and Weyl tensor, respectively. For the given tensor fields V and W, the notation VW represents some weighted sum of VW contractions with respect to the metric g(t) 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 R0=1. Since |Ric|(x,t)K and |p|(x,t)K2, (x,t)Bg(0)(x0,A)×[0,T], we haveeCK2tgi,j(x,0)gij(x,t)eCK2tgi,j(x,0). Consider the cut-off functionϕ(x):=(Adg(0)A)+, which is Lipschitz with suppϕ(x)Bg(0)(x0,A). From (38), it follows that|ϕ|g(t)eCK2T|ϕ|g(0)A1eCK2T. Apply Proposition 2.2 to getddtM|W|qϕ2q1KddtM|Ric|2|W|q1ϕ2qcK3ddtM|W|q1ϕ2q+cK3M|W|q1|ϕ|2ϕ2q2+cK3M|W|qϕ2q+cK5M|W|q1ϕ2q+cK5M|W|q2ϕ2q. By the Young inequality and the estimate (40), we haveddt(M|W|qϕ2q+1KM|R

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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