Related papers: Evolve nondegenerate two forms
In an earlier work joint with X. X. Chen and G. Tian, we introduced the weak K\"ahler-Ricci flow for various geometric motivations. In the current work, we take further consideration on setting up the weak flow. Namely, the initial class is…
We survey some recent developments on solutions of the K\"ahler-Ricci flow on compact K\"ahler manifolds which exist for all positive times.
We initiate the study of a new nonlinear parabolic equation on a Riemann surface. The evolution equation arises as a reduction of the Anomaly flow on a fibration. We obtain a criterion for long-time existence for this flow, and give a range…
This paper initiates a classification programme of flows of $\mathrm{SU}(2)$-structures on $4$-manifolds which have short-time existence and uniqueness. Our approach adapts a representation-theoretic method originally due to Bryant in the…
We exhibit the first examples of closed 4-manifolds with nonnegative sectional curvature that lose this property when evolved via Ricci flow.
We derive one unified formula for Ricci curvature tensor on arbitrary warped product manifold by introducing a new notation for the lift vector and the Levi-Civita connection.This formula is helpful to further consider Ricci flow (RF) and…
If we want to deform a compact Riemannian manifold with boundary using Ricci flow, we first need to decide on appropriate boundary conditions. We would like these conditions to reflect the geometric nature of the flow and allow for a…
This paper studies the long-time evolution of two point vortices under the 2D Navier-Stokes tokes equations. Starting from initial data given by a pair of Dirac measures, we derive an asymptotic expansion for the vorticity over time scales…
We consider complete (possibly non-compact) three dimensional Riemannian manifolds (M,g) such that: a) (M,g) is non-collapsed, b) the Ricci curvature of (M,g) is bounded from below, c) the geometry of (M,g) at infinity is not too extreme.…
In this short note, we give simple proof of the Ricci flow's local existence and uniqueness on closed Einstein manifolds. We suggest a new setting for studying the space of Riemannian metrics on a compact manifold.
We consider the evolution of a Hermitian metric on a compact complex manifold by its Chern-Ricci form. This is an evolution equation first studied by M. Gill, and coincides with the Kahler-Ricci flow if the initial metric is Kahler. We find…
In earlier work, we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities…
The Ricci flow is one of the most important topics in differential geometry, and a central focus of modern geometric analysis. In this paper, we give an illustrated introduction to the subject which is intended for a general audience. The…
We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and…
We are concerned with a model describing the motion of two compressible, immiscible fluids with density-dependent viscosity in the whole $\mathbb R^3$. The phases of the flow may have different pressure and viscosity laws and are separated…
Type II (ancient) solutions to the Ricci flow on surfaces are not yet classified. It is conjectured that the Rosenau solution and the cigar are the only solutions, modulo scaling. In this paper, we mainly study the backward limit and the…
A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of…
We continue studying a parabolic flow of almost K\"{a}hler structures introduced by Streets and Tian which naturally extends K\"{a}hler-Ricci flow onto symplectic manifolds. In the system of primarily the symplectic form, almost complex…
We consider the source-free Type IIA flow introduced by Fei-Phong-Picard-Zhang, and we study it in the case where the relevant geometric datum is a symplectic half-flat SU(3)-structure. We show the existence of ancient, immortal and eternal…
In the third part of this series of papers, we establish several topological results that will become important for studying the long-time behavior of Ricci flows with surgery. In the first part of this paper we recall some elementary…