相关论文: Entropy, noncollapsing, and a gap theorem for anci…
In our previous work we showed that for an ancient solution to the Ricci flow with nonnegative curvature operator, assuming bounded geometry on one time slice, bounded entropy implies noncollapsing on all scales. In this paper we prove the…
In [ZY2], the second author proved Perelman's assertion, namely, for an ancient Ricci flow with bounded and nonnegative curvature operator, bounded entropy is equivalent to noncollapsing on all scales. In this paper, we continue this…
In this paper, we study $\kappa$-noncollapsed ancient solutions to the Ricci flow with nonnegative curvature operator in higher dimensions. We impose one further assumption: one of the asymptotic shrinking gradient Ricci solitons is the…
In this paper, we study the classification of $\kappa$-noncollapsed ancient solutions to three-dimensional Ricci flow on $S^3$. We prove that such a solution is either isometric to a family of shrinking round spheres, or the Type II ancient…
In this paper we study the classification of compact $\kappa$-noncollapsed ancient solutions to the 3-dimensional Ricci flow which are rotationally and reflection symmetric. We prove that any such solution is isometric to the sphere or the…
In this survey we review Hamilton's entropy and Perelman's entropy, and provide motivations for these concepts. Then we review recent results on the logarithmic Sobolev inequality, the Sobolev inequalities and kappa-noncollapsing estimates…
In this paper, we study the classification of $\kappa$-noncollapsed ancient solutions to n-dimensional Ricci flow on $S^n$, extending the result in [13] to higher dimensions. We prove that such a solution is either isometric to a family of…
We give a complete classification of all $\kappa$-noncollapsed, complete ancient solutions to the K\"ahler Ricci flow with nonnegative bisectional curvature.
We consider compact ancient solutions to the three-dimensional Ricci flow which are noncollapsed. We prove that such a solutions is either a family of shrinking round spheres, or it has a unique asymptotic behavior as $t \to -\infty$ which…
We consider compact noncollapsed ancient solutions to the 3-dimensional Ricci flow that are rotationally and reflection symmetric. We prove that these solutions are either the spheres or they all have unique asymptotic behavior as…
It is known from work of Perelman that any finite-time singularity of the Ricci flow on a compact three-manifold is modeled on an ancient $\kappa$-solution. We prove that the every noncompact ancient $\kappa$-solution in dimension $3$ is…
We show that any manifold admitting a non-collapsed, ancient Ricci flow must have finite fundamental group. This generalizes what was known for $\kappa$-solutions in dimensions 2, 3. We furthermore show that this fundamental group must be a…
In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a…
$\mathscr{W}$-entropy and reduced volume for the Ricci flow were introduced by Perelman, which had proved their importance in the study of the Ricci flow. L. Ni studied the analogous concepts for the linear heat equation on the static…
Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in…
We study ancient Ricci flows which admit asymptotic solitons in the sense of Perelman. We prove that the asymptotic solitons must coincide with Bamler's tangent flows at infinity. Furthermore, we show that Perelman's $\nu$-functional is…
We introduce a new entropy functional for nonnegative solutions of the heat equation on a manifold with time-dependent Riemannian metric. Under certain integral assumptions, we show that this entropy is non-decreasing, and moreover convex…
We show that every $n$-dimensional, $\kappa$-noncollapsed, noncompact, complete ancient solution to the Ricci flow with uniformly PIC for $n=4$ or $n\ge 12$ has weakly PIC$_2$ and bounded curvature. Combining this with earlier results, we…
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 show that $\kappa$-solutions to the Ricci flow in dimensions $n\geq 4$ whose asymptotic shrinking Ricci soliton is the round cylinder $\mathbb{S}^{n-1}\times\mathbb{R}$ must be uniformly PIC. Combined with earlier classification results,…