Related papers: A Bochner Formula on Path Space for the Ricci Flow

In his seminal work \cite{Ri96}, Rivin characterized finite ideal polyhedra in three-dimensional hyperbolic space. However, the characterization of infinite ideal polyhedra, as proposed by Rivin, has remained a long-standing open problem.…

We introduce a new approach to prove the global existence and uniqueness of suitable weak solutions of the heat flow of harmonic mappings into CAT(0) metric spaces. Our method allows also to prove Lipschitz continuity in spatial variables…

For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the…

We prove that the Ricci flow on CP^n blown-up at one point starting with any rotationally symmetric Kahler metric must develop Type I singularities. In particular, if the total volume does not go to zero at the singular time, the parabolic…

This project serves to analyze the behavior of Ricci Flow in five dimensional manifolds. Ricci Flow was introduced by Richard Hamilton in 1982 and was an essential tool in proving the Geometrization and Poincare Conjectures. In general,…

In this article we prove an $\epsilon$-regularity theorem for non-collapsed Ricci flows, and use this to prove new estimates for singularity models of Fano K\"ahler-Ricci flows. In the course of our proof, we find a criterion for uniform…

In this paper, we show that starting from a geodesic ball $\overline{B_{r_0}}(0)$ in $\mathbb{H}^n$, for $n\geq3$, with prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class $[g_{\mathbb{S}^{n-1}}]$…

There are described equations for a pair comprising a Riemannian metric and a Killing field on a surface that contain as special cases the Einstein Weyl equations (in the sense of D. Calderbank) and a real version of a special case of the…

The Ricci flow equation of a conformally flat Riemannian metric on a closed 2-dimensional configuration space is analysed. It turns out to be equivalent to the classical Hamilton-Jacobi equation for a point particle subject to a potential…

We prove that if the Ricci tensor $\mathrm{Ric}$ of a geodesically complete Riemannian manifold $M$, endowed with the Riemannian distance $\mathsf{d}$ and the Riemannian measure $\mathfrak{m}$, is bounded from below by a continuous function…

Identifying any conformally round metric on the $2$-sphere with a unique cross section on the standard lightcone in the $3+1$-Minkowski spacetime, we gain a new perspective on $2d$-Ricci flow on topological spheres. It turns out that in…

In the paper, we construct conservative Markov processes corresponding to the martingale solutions to the stochastic heat equation on $\mathbb{R}^+$ or $\mathbb{R}$ with values in a general Riemannian maifold, which is only assumed to be…

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold $\mathscr M=M\times \mathbb S^N\times I$, whose dimension depends on a parameter $N$ unbounded from above. We construct sequences of…

We derive the entropy formula for the linear heat equaiton on complete Riemannian manifolds with nonnegative Ricci curvature. As applications, we study the relation between the value of entropy and the volume of balls of various scales. The…

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to…

We consider the Ricci flow $\frac{\partial}{\partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rm\geq 0, |Rm(p)|\to 0, ~as ~d(o,p)\to 0.$ We prove that the Ricci…

We consider the Kaehler-Ricci flow on complete finite-volume metrics that live on the complement of a divisor in a compact Kaehler manifold X. Assuming certain spatial asymptotics on the initial metric, we compute the singularity time in…

Ansatze are constructed under which the solutions of 11D supergravity must be stationary points of a parabolic flow on a Riemannian manifold $M^{10-p}$. This parabolic flow turns out to be the Ricci flow coupled to a scalar field, a…

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We…

For an immortal Ricci flow on an $m$-dimensional $(m\ge 3)$ closed manifold, we show the following convergence results: (1) if the curvature and diameter are uniformly bounded, then any unbounded sequence of time slices sub-converges to a…