Related papers: Ricci flows with unbounded curvature

We give a simple proof of an extension of the existence results of Ricci flow of G.Giesen and P.M.Topping [GiT1],[GiT2], on incomplete surfaces with bounded above Gauss curvature without using the difficult Shi's existence theorem of Ricci…

Consider the unnormalized Ricci flow $(g_{ij})_t = -2R_{ij}$ for $t\in [0,T)$, where $T < \infty$. Richard Hamilton showed that if the curvature operator is uniformly bounded under the flow for all times $t\in [0,T)$ then the solution can…

We prove that the Ricci flow g(t) starting at any metric on the euclidean space that is invariant by a transitive nilpotent Lie group N, can be obtained by solving an ODE for a curve of nilpotent Lie brackets. By using that this ODE is the…

In this paper we construct a Ricci de Turck flow on any incomplete Riemannian manifold with bounded curvature. The central property of the flow is that it stays uniformly equivalent to the initial incomplete Riemannian metric, and in that…

We consider Ricci flow on a closed surface with cone points. The main result is: given a (nonsmooth) cone metric g_0 over a closed surface there is a smooth Ricci flow g(t) defined for (0,T], with curvature unbounded above, such that g(t)…

Let $g(t)$ be a complete solution to the Ricci flow on a noncompact manifold such that $g(0)$ is Kahler. We prove that if $|Rm(g(t))|_{g(t)}\le a/t$ for some $a>0$, then $g(t)$ is Kahler for $t>0$. We prove that there is a constant $…

We show for a non homogeneous boundary value problem for the Ricci flow on the disk that when the initial metric has positive curvature and the boundary is convex then the initial metric is deformed, via the normalized flow and along…

This is a continuation of the research in [16]. Let $(\overline{M},g_{-1})$ be a closed geodesic $r_0$-ball in the hyperbolic space $(\mathbb{H}^n,g_{-1})$. Let $m\neq1$ be a positive constant. In this paper, we show that for $n\geq3$,…

In this paper we study $n$-dimensional Ricci flows $(M^n,g(t))_{t\in [0,T)},$ where $T< \infty$ is a potentially singular time, and for which the spatial $L^p$ norm, $p>\frac n 2$, of the scalar curvature is uniformly bounded on $[0,T).$ In…

We give a proof of the fact that the upper and the lower sectional curvature bounds of a complete manifold vary at a bounded rate under the Ricci flow.

Let M be a compact n-dimensional manifold, $n\ge 2$, with metric g(t) evolving by the Ricci flow $\partial g_{ij}/\partial t=-2R_{ij}$ in (0,T) for some $T\in\Bbb{R}^+\cup\{\infty\}$ with $g(0)=g_0$. Let $\lambda_0(g_0)$ be the first…

We establish a short-time existence theory for complete Ricci flows under scaling-invariant curvature bounds, starting from rotationally symmetric metrics on $\mathbb{R}^{n+1}$ that are noncollapsed at infinity, without assuming bounded…

We use a first-order energy quantity to prove a strengthened statement of uniqueness for the Ricci flow. One consequence of this statement is that if a complete solution on a noncompact manifold has uniformly bounded Ricci curvature, then…

In this paper, we study the asymptotic geometry of a noncollapsed ancient Ricci flow with nonnegative Ricci curvature via its tangent flow at infinity -- a noncollapsed $\mathbb{F}$-limit metric soliton [Bam23,CMZ23]. We first prove some…

In this paper, we prove a general maximum principle for the time dependent Lichnerowicz heat equation on symmetric tensors coupled with the Ricci flow on complete Riemannian manifolds. As an application we construct complete manifolds with…

In this paper we prove that for a given K\"ahler-Ricci flow with uniformly bounded Ricci curvatures in an arbitrary dimension, for every sequence of times $t_i$ converging to infinity, there exists a subsequence such that $(M,g(t_i + t))\to…

We prove that if the Ricci curvature is uniformly bounded under the Ricci-Harmonic flow for all times $t$ \in[0, T), then the curvature tensor has to be uniformly bounded as well.

We consider Riemannian manifolds $(M^n,g_0)$, $(M^n,h)$, where $(M^n,h)$ is smooth, complete, with curvature bounded in absolute value by $K_0 < \infty$, and $(1-\varepsilon_0(n)) h \leq g_0 \leq (1+\varepsilon_0(n)) h$ for some small…

We show that every complete non-compact three-manifold with non-negatively pinched Ricci curvature admits a complete Ricci flow solution for all positive time, with scale-invariant curvature decay and preservation of pinching. Combining…

In this paper we survey the recent developments of the Ricci flows on complete noncompact K\"{a}hler manifolds and their applications in geometry.