| In this article, firstly, we consider the monotonicity of eigenvalues of-Δ+S/2under a new geometric flow, that is, the Ricci-flow coupled with a Harmonic map. Next, similar to the discussion of "no breather" established by Perelman, we define some new energy functionals, i.e.,.F-energy functional and W-energy functional, and discuss their monotonicity and applications.In chapter2and chapter3, from the underlying equation λf=-Δf+2/1Sf and by identity transformation, items constructing and other fine computations, we get the monotonicity formulas of eigenvalue under the RHα-flow and normalized RHα-flow, re-spectively.In chapter4,5and6, By the construction of entropy functionals Fk and Wek and using differmorphism invariance principle(Corollary6.3.), we obtain the monotonicity formulas of Fk-functional and Wek-functional under the RHα-flow.Moreover, in chapter5and7, we investigate the applications of those monotonicity formulas, by employing differmorphism invariance principle and solving backward heat equation, we obtain the "no breather" property, i.e., a breather must be a gradient solition under the RHα-flow. |
