arXiv subjects

Man-Chun Lee

Publications and source records attributed to Man-Chun Lee.

63 records · Page 4Linked to original sources

arXiv2019

Instantaneously complete Chern-Ricci flow and Kähler-Einstein metrics

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 results can be viewed as generalization of an existence result by Giesen and Topping for surfaces of hyperbolic type of Ricci flow to higher dimensions in certain sense. On the other hand, we also discuss the long time behaviour of the solution and obtain some sufficient conditions for the existence of Kähler-Einstein metric on complete noncompact Hermitian manifolds, which generalizes the work of Lott-Zhang and Tosatti-Weinkove to complete noncompact Hermitian manifolds with possibly unbounded curvature.

math.DG↗

arXiv2019

The Kähler Ricci flow around complete bounded curvature Kähler metrics

We produce complete bounded curvature solutions to Kähler-Ricci flow with existence time estimates, assuming only that the initial data is a smooth \K metric uniformly equivalent to another complete bounded curvature \K metric. We obtain related flow results for non-smooth as well as degenerate initial conditions. We also obtain a stability result for complex space forms under the flow.

math.DG↗

arXiv2019

Complex manifolds with negative curvature operator

We prove that compact complex manifolds with admitting metrics with negative Chern curvature operator either admit a $dd^c$-exact positive (1,1) current, or are Kähler with ample canonical bundle. In the case of complex surfaces we obtain a complete classification. The proofs rely on a global existence and convergence result for the pluriclosed flow.

math.DG↗

arXiv2019

Uniqueness Theorem for non-compact mean curvature flow with possibly unbounded curvatures

In this paper, we discuss uniqueness and backward uniqueness for mean curvature flow of non-compact manifolds. We use an energy argument to prove two uniqueness theorems for mean curvature flow with possibly unbounded curvatures. These generalize the results by Chen and Yin. Using similar method, we also obtain a uniqueness result on Ricci flows. A backward uniqueness theorem is also proved for mean curvature flow with bounded curvatures.

math.DG↗

arXiv2018

Weakly PIC1 manifolds with maximal volume growth

In this article we use Ricci flow to show that complete PIC1 manifolds with maximal volume growth are diffeomorphic to $\mathbb{R}^n$. One of the key ingredients is local estimates of curvature lower bounds on an initial time interval of the Ricci flow. As another application of these estimates we obtain pseudolocality type results related to the PIC1 condition.

math.DG↗

arXiv2018

On the uniqueness of Ricci flow

In this note, we study the problem of uniqueness of Ricci flow on complete noncompact manifolds. We consider the class of solutions with curvature bounded above by C/t when t > 0. In paricular, we proved uniqueness if in addition the initial curvature is of polynomial growth and Ricci curvature of the flow is relatively small.

math.DG↗

arXiv2018

Longtime existence of Kähler Ricci flow and holomorphic sectional curvature

In this work, we obtain a existence criteria for the longtime Kähler Ricci flow solution. Using the existence result, we generalize a result by Wu-Yau on the existence of Kähler Einstein metric to the case with possibly unbounded curvature. Moreover, the Kähler Einstein metric with negative scalar curvture must be unique up to scaling.

math.DG↗

arXiv2017

Chern-Ricci flows on noncompact complex manifolds

In this work, we obtain existence criteria for Chern-Ricci flows on noncompact manifolds. We generalize a result by Tossati-Wienkove on Chern-Ricci flows to noncompact manifolds and at the same time generalize a result for Kahler-Ricci flows by Lott-Zhang to Chern-Ricci flows. Using the existence results, we prove that any complete noncollapsed Kahler metric with nonnegative bisectional curvature on a noncompact complex manifold can be deformed to a complete Kahler metric with nonnegative and bounded bisectional curvature which will have maximal volume growth if the initial metric has maximal volume. Combining this result with the result of Chau-Tam, we give another proof that a complete noncompact Kahler manifold with nonnegative bisectional curvature (not necessarily bounded) and maximal volume growth is biholomorphic to the complex Euclidean space. This last result has already been proved by Gang Liu recently using other methods. This last result is partial confirmation of a uniformization conjecture of Yau.

math.DG↗

arXiv2017

On existence and curvature estimates of Ricci flow

In this work, using the method by He, we prove a short time existence for Ricci flow on a complete noncompact Riemannian manifold with the following properties: (i) there is $r_0>0$ such that the volume of any geodesic balls of radius $r\le r_0$ is close to the volume of the geodesic ball in the Euclidean space with the same radius; (ii) the Ricci curvature is bounded below; (iii) the sectional curvature is bounded below by $-Cd^2(x,x_0)$ where $d(x,x_0)$ is the distance from a fixed point. An application to the uniformization for complete noncompact Kahler manifolds with nonnegative bisectional curvature and maximal volume growth is given. The result is related to the results of He and Liu. We prove if $g(t)$ is a complete solution to the Kahler-Ricci flow on a complete noncompact Kahler manifold so that $g(0)$ has nonnegative bisectional curvature and the curvature of $g(t)$ is bounded by $at^{-b}$ for some $a>0, 2>b>0$, then $g(t)$ also has nonnegative bisectional curvature. This generalizes a previous result. Under the assumption that $g(0)$ is noncollapsing, with additional assumption that the curvature of $g(t)$ is uniformly bounded away from $t>0$, we prove that the curvature of $g(t)$ is in fact bounded by $at^{-1}$ for some $a>0$. This result is proved using the method by Simon and Topping.

math.DG↗