scholarly journals The Kähler-Ricci flow, Ricci-flat metrics and collapsing limits

2018 ◽  
Vol 140 (3) ◽  
pp. 653-698 ◽  
Author(s):  
Valentino Tosatti ◽  
Ben Weinkove ◽  
Xiaokui Yang
Keyword(s):  
Ricci Flow ◽  
Ricci Flat

scholarly journals Brownian motion on Perelman’s almost Ricci-flat manifold

2020 ◽  
Vol 2020 (764) ◽  
pp. 217-239
Author(s):  
Esther Cabezas-Rivas ◽  
Robert Haslhofer
Keyword(s):  
Brownian Motion ◽  
Ricci Flow ◽  
Parallel Transport ◽  
Orthonormal Frame ◽  
Frame Bundle ◽  
Flat Manifold ◽  
Ricci Flat ◽  
Martingale Problems ◽  
Orthonormal Frame Bundle ◽  
The Laplace Operator

AbstractWe study Brownian motion and stochastic parallel transport on Perelman’s almost Ricci flat manifold {\mathcal{M}=M\times\mathbb{S}^{N}\times I}, whose dimension depends on a parameter N unbounded from above. We construct sequences of projected Brownian motions and stochastic parallel transports which for {N\to\infty} converge to the corresponding objects for the Ricci flow. In order to make precise this process of passing to the limit, we study the martingale problems for the Laplace operator on {\mathcal{M}} and for the horizontal Laplacian on the orthonormal frame bundle {\mathcal{OM}}. As an application, we see how the characterizations of two-sided bounds on the Ricci curvature established by A. Naber applied to Perelman’s manifold lead to the inequalities that characterize solutions of the Ricci flow discovered by Naber and the second author.

scholarly journals A C0-estimate for the parabolic Monge–Ampère equation on complete non-compact Kähler manifolds

2009 ◽  
Vol 146 (1) ◽  
pp. 259-270 ◽  
Author(s):  
Albert Chau ◽  
Luen-Fai Tam
Keyword(s):  
Ricci Flow ◽  
Kähler Manifolds ◽  
Main Step ◽  
Kahler Manifolds ◽  
Compact Sets ◽  
Ricci Flat ◽  
Long Time ◽  
Flat Manifolds ◽  
Monge Ampere ◽  
Invent Math

AbstractIn this article we study the Kähler–Ricci flow, the corresponding parabolic Monge–Ampère equation and complete non-compact Kähler–Ricci flat manifolds. Our main result states that if (M,g) is sufficiently close to being Kähler–Ricci flat in a suitable sense, then the Kähler–Ricci flow has a long time smooth solution g(t) converging smoothly uniformly on compact sets to a complete Kähler–Ricci flat metric on M. The main step is to obtain a uniform C0-estimate for the corresponding parabolic Monge–Ampère equation. Our results on this can be viewed as parabolic versions of the main results of Tian and Yau [Complete Kähler manifolds with zero Ricci curvature. II, Invent. Math. 106 (1990), 27–60] on the elliptic Monge–Ampère equation.

scholarly journals Stability of the Ricci flow at Ricci-flat metrics

2002 ◽  
Vol 10 (4) ◽  
pp. 741-777 ◽  
Author(s):  
Christine Guenther ◽  
James Isenberg ◽  
Dan Knopf
Keyword(s):  
Ricci Flow ◽  
Ricci Flat

scholarly journals Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

2020 ◽  
Author(s):  
Shaosai Huang ◽  
◽  
Xiaochun Rong ◽  
Bing Wang ◽  
◽  
...  
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Ricci Flow ◽  
Smoothing Techniques ◽  
Ricci Flat ◽  
Recent Developments ◽  
Open Question ◽  
Bounded Below

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.

Ricci Flow on Quasi Einstein Manifold

2010 ◽  
Vol 0 (-1) ◽  
pp. 447-454
Author(s):  
A. Bhattacharyya ◽  
T. De
Keyword(s):  
Ricci Flow ◽  
Einstein Manifold

scholarly journals Deformation classes in generalized Kähler geometry

2020 ◽  
Vol 7 (1) ◽  
pp. 241-256
Author(s):  
Matthew Gibson ◽  
Jeffrey Streets
Keyword(s):  
Symmetry Group ◽  
Ricci Flow ◽  
Kahler Geometry ◽  
Poisson Tensor ◽  
Natural Extensions ◽  
Generalized Kähler Geometry ◽  
Kähler Structures

AbstractWe describe natural deformation classes of generalized Kähler structures using the Courant symmetry group, which determine natural extensions of the notions of Kähler class and Kähler cone to generalized Kähler geometry. We show that the generalized Kähler-Ricci flow preserves this generalized Kähler cone, and the underlying real Poisson tensor.

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