H~k-Flow And Connection Ricci Flow

Geometric flow is an exciting and fruitful area of the modern differential ge-ometry. It means that the speed of the evolving manifold has some geometric interpretation, usually associated with some extrinsic or intrinsic curvature. The well-known examples are the Mean curvature flow whose speed function is the mean curvature vector of the submanifold, and the Ricci flow which evolves metric by Ricci curvature.In the case of mean curvature flow, Huisken [15] showed that the convex hyper-surfaces moving under such equations contract to points in finite time, and that the hypersurfaces become spherical in shape in the process. This argument has been extended to many processes where convex hypersurfaces move with speeds given by homogeneous degree one, [7], [8], and also other positive degrees of homogeneity [47], [7], [3], [5].In this paper, we consider the Hk-flow introduced by Schulze [23], whose speed function is the kth power of the mean curvature. According to the blow up rate, we can divide the singularities into two classes:Typeâ… and Typeâ…¡. With the help of our monotonicity formula, we can show that rescaled limit of Type I singularity must be a critical point of the functional and can be described by an elliptic equation. In the case of Type II singularity, we proceed the so called blow-up argument, showing the limit of blow-up sequence is translating soliton. Moreover, we give asymptotic expression of the rotationally symmetric translating soliton, and construct "Wing-like" solution proving the preservation of growth rate.The Ricci flow was introduced by Hamilton [33], in order to gain insight into the geometrization conjecture of Thurston. The following work by himself [34]-[37], Perelman [38], [39], [40] and others, have successfully proof this conjecture. On the other hand, it also give the physicists lots of inspiration, such as solving the c-theorem in Nonlinear sigma model [46]. For Asymptotically Euclidean manifold the [48] showed the preservation of total mass under the Ricci flow, while [49] studied the changes of the quasilocal mass.In this paper, we also generalize the idea of [48] to a larger class of Renormal-ization Group Flow, which can be consider as the Ricci flow on the manifold with torsion. We show the preservation of Asymptotically Euclidean as well as the ADM Mass.