# Kähler manifolds with almost nonnegative curvature

@article{Lee2021KhlerMW, title={K{\"a}hler manifolds with almost nonnegative curvature}, author={Man-Chun Lee and Luen-Fai Tam}, journal={Geometry \& Topology}, year={2021} }

In this paper, we construct local and global solutions to the Kahler-Ricci flow from a non-collapsed Kahler manifold with curvature bounded from below. Combines with the mollification technique of McLeod-Simon-Topping, we show that the Gromov-Hausdorff limit of sequence of complete noncompact non-collapsed Kahler manifolds with orthogonal bisectional curvature and Ricci curvature bounded from below is homeomorphic to a complex manifold. We also use it to study the complex structure of complete…

## 5 Citations

Comparison geometry of holomorphic bisectional curvature for Kähler manifolds and limit spaces

- MathematicsDuke Mathematical Journal
- 2021

We give an analog of triangle comparison for Kaehler manifolds with a lower bound on the holomorphic bisectional curvature. We show that the condition passes to noncollapsed Gromov-Hausdorff limits.…

Some local maximum principles along Ricci flows

- MathematicsCanadian Journal of Mathematics
- 2020

Abstract In this work, we obtain a local maximum principle along the Ricci flow
$g(t)$
under the condition that
$\mathrm {Ric}(g(t))\le {\alpha } t^{-1}$
for
$t>0$
for some constant
${\alpha…

Ricci-Deturck flow from rough metrics and applications

- Mathematics, Computer Science
- 2022

This work studies the short-time existence theory of Ricci-Deturck starting from rough metrics which are bi-Lipschitz and have small local scaling invariant gradient concentration to study several stability problems related to the scalar curvature.

Small curvature concentration and Ricci flow smoothing

- MathematicsJournal of Functional Analysis
- 2022

The asymptotic behaviour of Bergman kernels

- Mathematics
- 2021

We give a uniform L asymptotic expansion of Bergman kernel on the collection of n-dimensional polarized Kähler manifolds (M,ω,L, h) with Ricci lower bound −Λ and noncollapsing condition Vol (B1(x)) ≥…

## References

SHOWING 1-10 OF 67 REFERENCES

On the complex structure of K\"ahler manifolds with nonnegative curvature

- Mathematics
- 2005

We study the asymptotic behavior of the K\"ahler-Ricci flow on K\"ahler manifolds of nonnegative holomorphic bisectional curvature. Using these results we prove that a complete noncompact K\"ahler…

Gromov‐Hausdorff Limits of Kähler Manifolds with Ricci Curvature Bounded Below II

- MathematicsCommunications on Pure and Applied Mathematics
- 2020

We study noncollapsed Gromov‐Hausdorff limits of Kähler manifolds with Ricci curvature bounded below. Our main result is that each tangent cone is homeomorphic to a normal affine variety. This…

On complete noncompact Kähler manifolds with positive bisectional curvature

- Mathematics
- 2002

Abstract.We prove that a complete noncompact Kähler manifold Mnof positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of Cn and we show that…

Gromov‐Hausdorff Limits of Kähler Manifolds with Bisectional Curvature Lower Bound

- Mathematics
- 2015

Given a sequence of complete Kähler manifolds Min with bisectional curvature lower bound and noncollapsed volume, we prove that the pointed Gromov‐Hausdorff limit is homeomorphic to a normal complex…

Short-time existence of the Ricci flow on complete, non-collapsed $3$-manifolds with Ricci curvature bounded from below

- Mathematics
- 2016

We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually…

Kähler-Ricci shrinkers and ancient solutions with nonnegative orthogonal bisectional curvature

- Mathematics
- 2019

Chern–Ricci flows on noncompact complex manifolds

- Mathematics
- 2017

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…

Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces

- MathematicsGeometry & Topology
- 2017

We use Ricci flow to obtain a local bi-Holder correspondence between Ricci limit spaces in three dimensions and smooth manifolds. This is more than a complete resolution of the three-dimensional case…

Ricci flow on quasiprojective manifolds II

- Mathematics
- 2013

We study the Ricci flow on complete Kaehler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial…

The Ricci flow under almost non-negative curvature conditions

- MathematicsInventiones mathematicae
- 2019

We generalize most of the known Ricci flow invariant non-negative curvature conditions to less restrictive negative bounds that remain sufficiently controlled for a short time. As an illustration of…