Related papers: On the K\"ahler-Ricci flow on complex surfaces
In this note we provide a proof of the following: Any compact KRS with positive bisectional curvature is biholomorphic to the complex projective space. As a corollary, we obtain an alternative proof of the Frankel conjecture by using the…
In this work, we obtain a existence criteria for the longtime K\"ahler Ricci flow solution. Using the existence result, we generalize a result by Wu-Yau on the existence of K\"ahler Einstein metric to the case with possibly unbounded…
In this paper we survey some results on Ricci flowing non-smooth initial data. Among other things, we give a non-exhaustive list of various weak initial data which can be evolved with the Ricci flow. We also survey results which show that…
In this note, we define and study K\"ahler-Ricci flow with initial data not being smooth with some natural applications.
Motivated by the problem of finding constant scalar curvature K\"ahler metrics, we investigate a Ricci iteration sequence of Rubinstein that discretizes the pseudo-Calabi flow. While the long time existence of the flow is still an open…
For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the…
Let $({\M}, g(t))$ be a K\"ahler Ricci flow with positive first Chern class. We prove a uniform isoperimetric inequality for all time. In the process we also prove a Cheng-Yau type log gradient bound for positive harmonic functions on…
We study the uniqueness problem for the K\"ahler-Ricci flow with a conical initial condition. Given a complete gradient expanding K\"ahler-Ricci soliton on a non compact manifold with quadratic curvature decay, including its derivatives, we…
We show that any noncompact Riemann surface admits a complete Ricci flow g(t), t\in[0,\infty), which has unbounded curvature for all t\in[0,\infty).
We show that a complete Ricci flow of bounded curvature which begins from a manifold with a Ricci lower bound, local entropy bound, and small local scale-invariant integral curvature control will have global point-wise curvature control at…
We study rigidity on certain K\"ahler manifolds with nonnegative Ricci curvature. Among others things, we show that a complete noncompact K\"ahler surface with nonnegative Ricci curvature, Euclidean volume growth and quadratic curvature…
We develop interconnections between the complex normalizing flow for data drawn from Borel probability measures on the twofold realification of the complex manifold and a nonlinear flow nearly K\"ahler-Ricci. The complex normalizing flow…
We survey several problems concerning Riemannian manifolds with positive curvature of one form or another. We describe the PIC1 notion of positive curvature and argue that it is often the sharp notion of positive curvature to consider.…
We prove uniqueness of instantaneously complete Ricci flows on surfaces. We do not require any bounds of any form on the curvature or its growth at infinity, nor on the metric or its growth (other than that implied by instantaneous…
We investigate the metric behavior of the Kahler-Ricci flow on the Hirzebruch surfaces, assuming the initial metric is invariant under a maximal compact subgroup of the automorphism group. We show that, in the sense of Gromov-Hausdorff, the…
We extend the concept of singular Ricci flow by Kleiner and Lott from 3d compact manifolds to 3d complete manifolds with possibly unbounded curvature. As an application of the generalized singular Ricci flow, we show that for any 3d…
In this paper, we discuss diameter bound and Gromov-Hausdorff convergence of a twisted conical K\"ahler-Ricci flow on the total spaces of some holomorphic submersions. We also observe that, starting from a model conical K\"ahler metric with…
We define the Ricci curvature on simplicial complexes by modifying the definition of the Ricci curvature on graphs, and we prove the upper and lower bounds of the Ricci curvature. These properties are generalizations of previous studies.…
In this work, we obtain some existence results of Chern-Ricci Flows and the corresponding Potential Flows on complex manifolds with possibly incomplete initial data. We discuss the behaviour of the solution as $t\rightarrow 0$. These…
The Ricci iteration is a discrete analogue of the Ricci flow. According to Perelman, the Ricci flow converges to a Kahler-Einstein metric whenever one exists, and it has been conjectured that the Ricci iteration should behave similarly.…