Related papers: Smooth Structures and Normalized Ricci Flows on No…

As a step toward understanding the analytic behavior of Type-III Ricci flow singularities, i.e. immortal solutions that exhibit |Rm|<C/t curvature decay, we examine the linearization of an equivalent flow at fixed points discovered recently…

In this paper we construct solutions to Ricci DeTurck flow in four dimensions on closed manifolds which are instantaneously smooth but whose initial values $g$ are (possibly) non-smooth Riemannian metrics whose components in smooth…

We show that the combination of non-negative sectional curvature (or $2$-intermediate Ricci curvature) and strict positivity of scalar curvature forces rigidity of complete (non-compact) two-sided stable minimal hypersurfaces in a…

We derive modified Perelman-type monotonicity formulas for solutions to the generalized Ricci flow equation with symmetry on principal bundles, which lead to rigidity and classification results for nonsingular solutions.

Let $(M^n,g_0)$ ($n$ odd) be a compact Riemannian manifold with $\lambda(g_0)>0$, where $\lambda(g_0)$ is the first eigenvalue of the operator $-4\Delta_{g_0}+R(g_0)$, and $R(g_0)$ is the scalar curvature of $(M^n,g_0)$. Assume the maximal…

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…

For any flag manifold $M=G/K$ of a compact simple Lie group $G$ we describe non-collapsing ancient invariant solutions of the homogeneous unnormalized Ricci flow. Such solutions emerge from an invariant Einstein metric on $M$, and by…

In this short note we show that non-negative Ricci curvature is not preserved under Ricci flow for closed manifolds of dimensions four and above, strengthening a previous result of Knopf in \cite{K} for complete non-compact manifolds of…

Understanding the relationships between geometry and topology is a central theme in Riemannian geometry. We establish two results on the fundamental groups of open (complete and noncompact) $n$-manifolds with nonnegative Ricci curvature and…

In each dimension $N\geq 3$ and for each real number $\lambda\geq 1$, we construct a family of complete rotationally symmetric solutions to Ricci flow on $\mathbb{R}^{N}$ which encounter a global singularity at a finite time $T$. The…

We introduce certain spherically symmetric singular Ricci solitons and study their stability under the Ricci flow from a dynamical PDE point of view. The solitons in question exist for all dimensions $n+1\ge 3$, and all have a point…

In this work, we prove uniqueness for complete non-compact Ricci flow with scaling invariant curvature bound. This generalizes the earlier work of Chen-Zhu, Kotschwar and covers most of the example of Ricci flows with unbounded curvature.…

In this paper, we study the singularities of two extended Ricci flow systems --- connection Ricci flow and Ricci harmonic flow using newly-defined curvature quantities. Specifically, we give the definition of three types of singularities…

In this work, we extend the existence theory of non-collapsed Ricci flows from point-wise curvature lower bound to Kato-type lower bound. As an application, we prove that compact three dimensional non-collapsed strong Kato limit space is…

We explicitly describe the solution of the G$_2$-Laplacian flow starting from an extremally Ricci-pinched closed G$_2$-structure on a compact 7-manifold and we investigate its properties. In particular, we show that the solution exists for…

In this paper, we study 4-dimensional complete non-compact manifold with its curvature operator in $\mathfrak{C}_{\eta,\mu}$ via Ricci flow. We obtain topological and geometric gap theorems assuming such manifold has maximal volume growth.…

We investigate the topological regularity and stability of noncollapsed Ricci limit spaces $(M_i^n,g_i,p_i)\to (X^n,d)$. We confirm a conjecture proposed by Colding and Naber in dimension $n=4$, showing that the cross-sections of tangent…

In this paper we prove convergence and compactness results for Ricci flows with bounded scalar curvature and entropy. More specifically, we show that Ricci flows with bounded scalar curvature converge smoothly away from a singular set of…

We proved that on every Stiefel manifold $V_2\mathbb{R}^n\cong \operatorname{SO}(n)/\operatorname{SO}(n-2)$ with $n\ge 3$ the normalized Ricci flow preserves the positivity of the Ricci curvature of invariant Riemannian metrics with…

We prove short time existence for the Ricci flow on open manifolds of nonnegative complex sectional curvature. We do not require upper curvature bounds. By considering the doubling of convex sets contained in a Cheeger-Gromoll convex…