The Ricci-Bourguignon flow (Q521738)

Let \(M\) be a compact smooth \(n\)-dimensional Riemannian manifold without boundary with an evolving metric \(g\left (t\right )\) on it satisfying the Ricci-Bourguignon flow, i.e., the quasilinear parabolic PDE \(\frac{ \partial g}{ \partial t} = -2\left (*\text{Ric} -\rho Rg\right )\) where \(*\text{Ric}\) and \(R\) denote the Ricci tensor and the scalar curvature on \(M\), respectively. For a special value of \(\rho \), the right-hand side of the Ricci-Bourguignon flow equation reduces to the Einstein tensor (\(\rho =1/2\)), the traceless Ricci tensor (\(\rho =1/n\)), the Schouten tensor (\(\rho =1/2(n -1)\)) and the Ricci tensor (\(\rho =0\)). First, the authors observe that if \(\rho >1/2\left (n -1\right )\), then there are no (even short-time) solutions of the Ricci-Bourguignon flow for general initial metric tensor data \(g_{0}\). The main result of Section 2 (Theorem 2.1) states that if \(\rho <1/2\left (n -1\right )\), then for any initial metric \(g_{0}\) on \(M\), there is a unique solution \(g(t)\) of the Ricci-Bourguignon flow equation for a positive time interval. The case when \(\rho =1/2\left (n -1\right )\) (Schouten flow) remains open, when \(n \geq 3\). In Section 3, the authors derive the evolution equations for the curvature. In Section 4, they employ results of Section 3 on the curvature evolution and the maximum principle to obtain conditions on the curvature which are invariant under the Ricci-Bourguignon flow. Results of Section 4 enable the authors to prove the Hamilton-Ivey estimate for \(n =3\). In Section 5, the authors prove a priori estimates (assuming that \(\rho <1/2(n -1)\)) on the Riemann tensor, when \(t \in \left [0 ,T\right ]\) (Theorem 5.6) and establish that if \(g\left (t\right )\) is a compact solution of the Ricci-Bourguignon flow equation on a finite maximal time interval \([0 ,T)\), then the Riemann tensor is unbounded as \(t\) approaches to \(T\) (Theorem 5.7).