On The Curvature Estimates For The Ricci Bourguignon Flow And The Conformal Ricci Flow

Geometric flow means that the speed of evolving manifold has some geometric interpretation,usually associated with some extrinstic or intrinsic curvature.The most well-known example is the Ricci flow introduced by Hamilton which is an important geometric and analytic tool in differential geometry and was studied extensively.In this paper,we study the extension problem of the Ricci Bourguignon flow on Riemannian manifolds as well as the curvature estimates of the conformal Ricci flow on Riemannian manifolds.In Chapter 2,we show that the norm of the Weyl tensor of any smooth solution to the Ricci Bourguignon flow can be explicitly estimated in terms of its initial value on a given ball,a local uniform bound on the Ricci tensor.As an application,we show that along the Ricci Bourguignon flow,if the Ricci curvature is bounded,then the curvature operator is bounded.In Chapter 3,we show that the norm of the Weyl tensor of any smooth solution to the conformal Ricci flow can be explicitly estimated in terms of its initial value on a given ball,a local uniform bound on the Ricci tensor and the potential function.On compact manifolds,the curvature operator is bounded if the Ricci curvature is bounded.