Related papers: A remark on Ricci Breathers and Solitons

In this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat,…

We formulate and solve the existence problem for Ricci flow on a Riemann surface with initial data given by a Radon measure as volume measure. The theory leads us to a large class of new examples of nongradient expanding Ricci solitons,…

In this short note, we show that homogeneous Ricci solitons are algebraic. As an application, we see that the generalized Alekseevskii conjecture is equivalent to the Alekseevskii conjecture.

Motivated by the Hamilton's Ricci flow, we define the homogeneous flow of a parallelizable manifold and show the long time existence and uniqueness of its solutions on $[0,\infty).$ Using this flow, we outline a simple proof of the Poincare…

We present in this paper a general approach to study the Ricci flow on homogeneous manifolds. Our main tool is a dynamical system defined on a subset H(q,n) of the variety of (q+n)-dimensional Lie algebras, parameterizing the space of all…

This paper investigates the short-time existence and uniqueness of Ricci flow solutions on Finsler manifolds. The main results of this paper are theorems demonstrating the short-time existence of the flow solution for $n$-dimensional…

We study the Ricci flow on $\mathbb{R}^{4}$ starting at an SU(2)-cohomogeneity 1 metric $g_{0}$ whose restriction to any hypersphere is a Berger metric. We prove that if $g_{0}$ has no necks and is bounded by a cylinder, then the solution…

In this note, we provide a very simple proof of the uniformization theorem of Riemann surfaces by Ricci flow. The argument builds on a refinement of Hamilton's isoperimetric estimate for the Ricci flow on the two-sphere.

We study the Ricci flow on Riemannian groupoids. We assume that these groupoids are closed and that the space of orbits is compact and connected. We prove the short time existence and uniqueness of the Ricci flow on these groupoids. We also…

This is the second paper in a series of works devoted to nonholonomic Ricci flows. By imposing non-integrable (nonholonomic) constraints on the Ricci flows of Riemannian metrics we can model mutual transforms of generalized Finsler-Lagrange…

In this short notes, we discuss monotonicity formulas under various rescaled versions of Ricci flow. The main result is Theorem \ref{theo rescaled}.

We prove that a complete solution to the Ricci flow on $M\times [-T, 0)$ which has quadratic curvature decay on some end of $M$ and converges locally smoothly to the end of a cone on that neighborhood as $t\nearrow 0$ must be a gradient…

We use analytical and numerical methods to investigate the equations for cohomogeneity one shrinking gradient Ricci solitons. We show the existence of a winding number for this system around the subvariety of phase space corresponding to…

Let $(M,g,\phi)$ be a solution to the Ricci flow coupled with the heat equation for a scalar field $\phi$. We show that a complete, $\kappa$-noncollapsed solution $(M,g,\phi)$ to this coupled Ricci flow with a Type I singularity at time…

Suppose $G$ is a compact Lie group, $H$ is a closed subgroup of $G$, and the homogeneous space $G/H$ is connected. The paper investigates the Ricci flow on a manifold $M$ diffeomorphic to $[0,1]\times G/H$. First, we prove a short-time…

As an application of his entropy formula, Perelman proved that every compact shrinking breather is a shrinking gradient Ricci soliton. We give a proof for the complete noncompact case by using Perelman's $\mathcal{L}$-geometry. Our proof…

We prove some results for the solitons of the Ricci-Bourguignon flow, generalizing corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci-Bourguignon $almost$ solitons…

We find exact solutions describing Ricci flows of four dimensional pp-waves nonlinearly deformed by two/three dimensional solitons. Such solutions are parametrized by five dimensional metrics with generic off-diagonal terms and connections…

I discuss certain applications of the Ricci flow in physics. I first review how it arises in the renormalization group (RG) flow of a nonlinear sigma model. I then review the concept of a Ricci soliton and recall how a soliton was used to…

In this note, we construct families of functionals of the type of $\mathcal{F}$-functional and $\mathcal{W}$-functional of Perelman. We prove that these new functionals are nondecreasing under the Ricci flow. As applications, we give a…