相关论文: Eternal Solutions to the Ricci Flow on $\R^2$

Ancient solutions arise in the study of Ricci flow singularities. Motivated by the work of Fateev on 3-dimensional ancient solutions we construct high dimensional ancient solutions to Ricci flow on spheres and complex projective spaces as…

New results are obtained for global regularity and long-time behavior of the solutions to the 2D Boussinesq equations for the flow of an incompressible fluid with positive viscosity and zero diffusivity in a smooth bounded domain. Our first…

We show that a simply-connected closed four-dimensional Ricci flow whose Ricci curvature is uniformly bounded below and whose volume does not approach zero must converge to a $C^{0}$ orbifold at any finite-time singularity, so has an…

We study noncommutative Ricci flow in a finite dimensional representation of a noncommutative torus. It is shown that the flow exists and converges to the flat metric. We also consider the evolution of entropy and a definition of scalar…

Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton…

Let $(M^3,g_0)$ be a complete noncompact Riemannian 3-manifold with nonnegative Ricci curvature and with injectivity radius bounded away from zero. Suppose that the scalar curvature $R(x)\to 0$ as $x\to \infty$. Then the Ricci flow with…

We produce complete bounded curvature solutions to K\"ahler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain…

We prove that a steady gradient Ricci soliton is either Ricci flat with a constant potential function or a quotient of the product steady soliton $N^{n-1}\times\mathbb{R}$, where $N^{n-1}$ is Ricci flat, or isometric to the Bryant soliton…

We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex…

We study Ricci flows on $R^n$, $n\ge 3$, that evolve from asymptotically flat initial data. Under mild conditions on the initial data, we show that the flow exists and remains asymptotically flat for an interval of time. The mass is…

In this article, we consider the space-time fractional (nonlocal) equation characterizing the so-called "double-scale" anomalous diffusion $$\partial_t^\beta u(t, x) = -(-\Delta)^{\alpha/2}u(t,x) - (-\Delta)^{\gamma/2}u(t,x) \ \ t> 0, \…

For all dimensions $n\geq5$, let $(M,g,f)$ be a $n-$dimensional shrinking gradient Ricci soliton with strictly positive isotropic curvature (PIC). Suppose furthermore that $\nabla^2f$ is $2-$nonnegative and the curvature tensor is WPIC1 at…

We extend the second part of \cite{Bre20} on the uniqueness of ancient $\kappa$-solutions to higher dimensions. In dimensions $n \geq 4$, an ancient $\kappa$-solution is a nonflat, complete, ancient solution of the Ricci flow that is…

We consider a modified Ricci flow equation whose stationary solutions include Einstein and Ricci soliton metrics, and we study the linear stability of those solutions relative to the flow. After deriving various criteria that imply linear…

We prove the growth rate of global solutions of the equation $u_t=\Delta u-u^{-\nu}$ in $\R^n\times (0,\infty)$, $u(x,0)=u_0>0$ in $\R^n$, where $\nu>0$ is a constant. More precisely for any $0<u_0\in C(\R^n)$ satisfying…

In this paper, we extend the theory of Ricci flows satisfying a Type-I scalar curvature condition at a finite-time singularity. In [Bam16], Bamler showed that a Type-I rescaling procedure will produce a singular shrinking gradient Ricci…

We introduce a notion of Ricci flow in generalized geometry, extending a previous definition by Gualtieri on exact Courant algebroids. Special stationary points of the flow are given by solutions to first-order differential equations, the…

We define several notions of singular set for Type I Ricci flows and show that they all coincide. In order to do this, we prove that blow-ups around singular points converge to nontrivial gradient shrinking solitons, thus extending work of…

We show that the set of awesome homogeneous metrics on non-compact manifolds is Ricci flow invariant. Moreover, if the universal cover of such awesome homogeneous space is not contractible the Ricci flow has finite extinction time,…

We study the short-time existence and regularity of solutions to a boundary value problem for the Ricci-DeTurck equation on a manifold with boundary. Using this, we prove the short-time existence and uniqueness of the Ricci flow prescribing…