Ricci-mean Curvature Flow And Connection Ricci Flow

Geometric curvature flow means the speed of the evolving manifold has some geometric interpretation, usually associated with some extrinsic or intrinsic cur-vature. The most classical ones seem to be Ricci flow and mean curvature flow for hypersurfaces. As the main tool for Perelman to solve the Poincare Conjec-ture, Ricci flow was introduced by Hamilton. Inspired by Hamilton’s Ricci flow, Huisken used the parabolic differential equation to consider mean curvature flow of hypersurfaces.In this paper, we first study Ricci-mean curvature flow. Ricci-mean curvature flow means a parameter of immersions X(·, t):Mnâ†'(Nn+1,g(t)), where X(-,t) evolving under the mean curvature flow while g(t) satisfying Ricci flow. In this case the behaviour of hypersurfaces also depends on the deformation of g(t). We focus on the convergence problem about Ricci-mean curvature flow. We prove that if the initial background manifold is an approximation of a spherical space form and the initial hypersurface also satisfies a suitable pinching condition, then either the hypersurfaces shrink to a round point at finite time or converge to a totally geodesic sphere as the time tends to infinity.By constructing the F-functional, Huisken showed that for mean curvature flow of closed hypersurface, the type I singularity corresponds to the self-shrinker. The recent breakthrough in the study of self-shrinker is due to the work of Colding and Minicozzi. They introduced the concept of entropy-stability for self-shrinker and gave the classification of entropy-stable self-shrinkers. For Ricci-mean curvature flow, if the ambient manifold is a gradient Ricci soliton, Magni-Mantegazza-Tsatis can construct a functional which is monotonic decreasing under Ricci-mean cur-vature. In particular, under the assumption that the ambient space is a gradient shrinking soltion (N,g,f), Yamamoto showed the Type I singularity corresponds to f-minimal hypersurface, i.e., the limiting hypersurface satisfies H= g(â–½f,v). Thus, it’s a natural question to classify f-minimal hypersurfaces. We consider the f-minimal hypersurfaces in Mn×R, where Mn is an Einstein manifold with positive Ricci curvature. By introducing a functional similar to Huisken’s F-functional, we give a geometric classification of f-minimal hypersurfaces.Renormalization group flow was introduced by physicists when investigating the nonlinear σ model. Inspired by that, Streets studied connection Ricci flow, i.e., a generalization of Ricci flow to connection with torsion. We consider connection Ricci flow for 3-dimensional closed manifolds. Singularity analysis plays a key role in the research of geometric curvature flow, and the classification of singularity usually depends on the blow up rate of the geometric quantities. By using evolution equations, we give a lower bound for the blow up rate of the curvature defined by the connection with torsion. Based on that, we classify the maximal solution of 3-dimension connection Ricci flow and show it corresponds to the relative singularity model under some reasonable condition.