Rotational symmetry of self-similar solutions to the Ricci flow

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman’s first paper.     

Ricci Yang-Mills flow on surfaces

We study the behavior of the Ricci Yang-Mills flow for U(1) bundles on surfaces. By exploiting a coupling of the Liouville and Yang-Mills energies we show that existence for the flow reduces to a bound on the isoperimetric constant or the L⁴ norm of the bundle curvature. We furthermore completely describe the behavior of long time solutions of this flow on surfaces. Finally, in Appendix A we classify all gradient solitons of this flow on surfaces.     

Normalized Ricci flow on nonparabolic surfaces

This paper studies the normalized Ricci flow on a nonparabolic surface, whose scalar curvature is asymptotically –1 in an integral sense. By a method initiated by R. Hamilton, the flow is shown to converge to a metric of constant scalar curvature –1. A relative estimate of Green’s function is proved as a tool.      

Ricci flow on surfaces with cusps

We consider the normalized Ricci flow ? _t g = (ρ ? R)g with initial condition a complete metric g ₀ on an open surface M where M is conformal to a punctured compact Riemann surface and g ₀ has ends which are asymptotic to hyperbolic cusps. We prove that when χ(M) < 0 and ρ < 0, the flow g(t) converges exponentially to the unique complete metric of constant Gauss curvature ρ/2 in the conformal class.     

Curvature integrals under the Ricci flow on surfaces

In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.      

Ricci flow on Kähler-Einstein surfaces

In this paper, we prove that if M is a K?hler-Einstein surface with positive scalar curvature, if the initial metric has nonnegative sectional curvature, and the curvature is positive somewhere, then the K?hler-Ricci flow converges to a K?hler-Einstein metric with constant bisectional curvature. In a subsequent paper [7], we prove the same result for general K?hler-Einstein manifolds in all dimension. This gives an affirmative answer to a long standing problem in K?hler Ricci flow: On a compact K?hler-Einstein manifold, does the K?hler-Ricci flow converge to a K?hler-Einstein metric if the initial metric has a positive bisectional curvature? Our main method is to find a set of new functionals which are essentially decreasing under the K?hler Ricci flow while they have uniform lower bounds. This property gives the crucial estimate we need to tackle this problem. Oblatum 8-IX-2000 & 30-VII-2001?Published online: 19 November 2001     

Ricci flow of negatively curved incomplete surfaces

We show uniqueness of Ricci flows starting at a surface of uniformly negative curvature, with the assumption that the flows become complete instantaneously. Together with the more general existence result proved in [10], this settles the issue of well-posedness in this class.     

Non-singular solutions to the normalized Ricci flow equation

In this paper, we study non-singular solutions to Ricci flow on a closed manifold of dimension at least 4. Amongst other things we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t > 0 with uniformly bounded sectional curvature, then the Euler characteristic . Moreover, the 4-manifold satisfies one of the followings

A note on uniformization of Riemann surfaces by Ricci flow

We clarify that the Ricci flow can be used to give an independent proof of the uniformization theorem of Riemann surfaces.

     

On the long-time behavior of type-III Ricci flow solutions

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton’s compactness theorem that does not assume a lower injectivity radius bound, in terms of Riemannian groupoids. Using this, we show that the long-time behavior of type-III Ricci flow solutions is governed by the dynamics of an -action on a compact space. This work was supported by NSF grant DMS-0306242     

The conjugate heat equation and Ancient solutions of the Ricci flow

We prove Gaussian type bounds for the fundamental solution of the conjugate heat equation evolving under the Ricci flow. As a consequence, for dimension 4 and higher, we show that the backward limit of Type I κ-solutions of the Ricci flow must be a non-flat gradient shrinking Ricci soliton. This extends Perelman?s previous result on backward limits of κ-solutions in dimension 3, in which case the curvature operator is nonnegative (it follows from Hamilton–Ivey curvature pinching estimate). As an application, this also addresses an issue left in Naber (2010) [23], where Naber proves the interesting result that there exists a Type I dilation limit that converges to a gradient shrinking Ricci soliton, but that soliton might be flat. The Gaussian bounds that we obtain on the fundamental solution of the conjugate heat equation under evolving metric might be of independent interest.     

Collapsing sequences of solutions to the Ricci flow on 3-manifolds with almost nonnegative curvature

We study sequences of 3-dimensional solutions to the Ricci flow with almost nonnegative sectional curvatures and diameters tending to infinity. Such sequences may arise from the limits of dilations about singularities of Type IIb. In particular, we study the case when the sequence collapses, which may occur when dilating about infinite time singularities. In this case we classify the possible Gromov-Hausdorff limits and construct 2-dimensional virtual limits. The virtual limits are constructed using Fukaya theory of the limits of local covers. We then show that the virtual limit arising from appropriate dilations of a Type IIb singularity is always Hamilton's cigar soliton solution. Partially supported by NSF grant DMS-0203926.     

Rotational symmetry of the Hermite projection operators

We calculate an integral formula for the Hermite projection operators. We give some applications of our formula. We also give a short proof of a recent theorem of Thangavelu.

     

Immortal solution of the Ricci flow

For any complete noncompact Kahler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow.     

Immortal solution of the Ricci flow

For any complete noncompact Kähler manifold with nonnegative and bounded holomorphic bisectional curvature, we provide the necessary and sufficient condition for the immortal solution to the Ricci flow.