Reconciliation RICCI SOLITION

The Ricci flow developed by Hamilton since the1980s has made significant progress and obtained many important results, including the proof of the famous Poincare Conjecture. The purpose of this article is to use the Ricci flow theory to analysis soliton-important self-similar solution of harmonic Ricci flow, which has a extensive physical background. The harmonic Ricci flow is defined by the following system of nonlinear partial differential equations:Where g andφ are complete metric and function on the manifold, an is a constant depending only on the dimension of the manifold. A uniform Rimann-bound is enough to conclude aφ-bound and then existence of the solution. The coupled system may behave less singular than the Ricci flow or the standard harmonic map flow alone.In this article, we first summarize the existing results about the harmonic Ricci flow and give some different proofs, including that we use the maximum principle to deduce that if (g(t),φ(t) is a steady or expanding soliton of the harmonic Ricci flow on a compact manifold, then φ must be a constant and the metric g(t) is corresponding Ricci soliton; similar to Perelman’s F and W functional in Ricci flow, through the variational of the F and W functionals in harmonic Ricci flow, we prove that breather must be corresponding gradient soliton. At last, using the compactness theorem by Berhard List,we get the singularity models and discuss the relationship between singularity models and soltions.