Ricci flow papers from 2017

We consider an axisymmetric closed hypersurface evolving by its mean curvature with driving force under singular initial hypersurface. We study this problem by level set method. We give some criteria to judge whether the interface evolution is fattening or non-fattening.

The long-time existence and umbilicity estimates for compact, graphical solutions to expanding curvature flows are deduced in Riemannian warped products of a real interval with a compact fibre. Notably we do not assume the ambient manifold to be rotationally symmetric, nor the radial curvature to converge, nor a lower bound on the ambient sectional curvature. The inverse speeds are given by powers $p\le 1$ of a curvature function satisfying few common properties.

We study Brownian motion and stochastic parallel transport on Perelman's almost Ricci flat manifold $\mathcal{M}=M\times {\mathbb{S}}^N\times I$, whose dimension depends on a parameter $N$ unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for $N\to \infty$ converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on $\mathcal{M}$ and for the horizontal Laplacian on the orthonormal frame bundle $\mathcal{OM}$ . As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman's manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

In this paper, we introduce a class of Sasaki manifolds with a reductive $G$-group action, called $G$-Sasaki manifolds. By reducing K-energy to a functional defined on a class of convex functions on a moment polytope, we give a criterion for the properness of K-energy. In particular, we deduce a sufficient and necessary condition related to the polytope for the existence of transverse $G$-Sasaki Einstein metrics. A similar result is also obtained for transverse $G$-Sasaki Ricci solitons. As an application, we construct several examples of $G$-Sasaki Ricci solitons by an established openness theorem for transverse $G$-Sasaki Ricci solitons.

In this note we study a large class of mean curvature type flows of graphs in product manifold $N\times R$ where N is a closed Riemann- ian manifold. Their speeds are the mean curvature of graphs plus a prescribed function. We establish long time existence and uniformly convergence of those flows with a barrier condition and a condition on the derivative of prescribed function with respect to the height. As an application we construct a weighted mean curvature flow in large classes of warped product manifolds which evolves each graph into a totally ge- odesic slice

We consider the CR Yamabe flow on a compact strictly pseudoconvex CR manifold $M$ of real dimension $2n+1$. We prove convergence of the CR Yamabe flow when $n=1$ or $M$ is spherical.

We complete the proof of the Generalized Smale Conjecture, apart from the case of $R{P}^3$, and give a new proof of Gabai's theorem for hyperbolic 3-manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than $S^3$ and $R{P}^3$ and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold $X$, the inclusion $\text{Isom}(X,g)\to \text{Diff}(X)$ is a homotopy equivalence for any Riemannian metric $g$ of constant sectional curvature.

English translation of "Solitony Ricciego" (Wiadomości Matematyczne 48, 2012, no. 1, pp. 1-32). Despite the general-sounding title, the text covers just a few narrow topics: Perelman's proof of the fact that compact Ricci solitons are of the gradient type, and a detailed unified description of Page's and Berard Bergery's Einstein manifolds on the one hand, and Koiso's and Cao's Kähler-Ricci solitons on the other, as the Ricci solitons arising via conformal changes from certain special Kähler manifolds.

In this paper, we consider the area-preserving mean curvature flow with free Neumann boundaries. We show that for a rotationally symmetric $n$-dimensional hypersurface in ${\mathbb{R}}^{n+1}$ between two parallel hyperplanes will converge to a cylinder with the same area under this flow. We use the geometric properties and the maximal principle to obtain gradient and curvature estimates, leading to long-time existence of the flow and convergence to a constant mean curvature surface.

We prove a compactness theorem for Kähler metrics with various bounds on Ricci curvature and the $\mathcal{I}$ functional. We explore applications of our result to the continuity method and the Calabi flow.

We show that the space of asymptotically conical self-expanders of the mean curvature flow is a smooth Banach manifold. An immediate consequence is that non-degenerate self-expanders – that is, those self-expanders that admit no non-trivial normal Jacobi fields that fix the asymptotic cone – are generic in a certain sense.

We prove that a shrinking gradient Ricci soliton which agrees to infinite order at spatial infinity with one of the standard cylindrical metrics on $S^k\times {\text{RR}}^{n-k}$ for $k\ge 2$ along some end must be isometric to the cylinder on that end. When the underlying manifold is complete, it must be globally isometric either to the cylinder or (when $k=n-1$) to its ${\text{ZZ}}_2$-quotient.