scholarly journals Normalized Ricci flows and conformally compact Einstein metrics

2011 ◽  
Vol 46 (1-2) ◽  
pp. 183-211 ◽  
Author(s):  
Jie Qing ◽  
Yuguang Shi ◽  
Jie Wu
Keyword(s):  
Einstein Metrics ◽  
Conformally Compact ◽  
Ricci Flows

NORMALIZED RICCI FLOW, SURGERY, AND SEIBERG–WITTEN INVARIANTS

2014 ◽  
Vol 25 (02) ◽  
pp. 1450005
Author(s):  
MASASHI ISHIDA
Keyword(s):  
Existence Theorem ◽  
Euler Characteristic ◽  
Ricci Flow ◽  
Einstein Metrics ◽  
Nonsingular Solution ◽  
Ricci Flows ◽  
Normalized Ricci Flow ◽  
Four Manifolds ◽  
Special Case

We investigate the behavior of solutions of the normalized Ricci flow under surgeries of four-manifolds along circles by using Seiberg–Witten invariants. As a by-product, we prove that any pair (α, β) of integers satisfying α + β ≡ 0 (mod 2) can be realized as the Euler characteristic χ and signature τ of infinitely many closed smooth 4-manifolds with negative Perelman's [Formula: see text] invariants and on which there is no nonsingular solution to the normalized Ricci flows for any initial metric. In particular, this includes the existence theorem of non-Einstein 4-manifolds due to Sambusetti [An obstruction to the existence of Einstein metrics on 4-manifolds, Math. Ann.311 (1998) 533–547] as a special case.

scholarly journals Boundary regularity of conformally compact Einstein metrics

2005 ◽  
Vol 69 (1) ◽  
pp. 111-136 ◽  
Author(s):  
Piotr T. Chruściel ◽  
Erwann Delay ◽  
John M. Lee ◽  
Dale N. Skinner
Keyword(s):  
Einstein Metrics ◽  
Boundary Regularity ◽  
Conformally Compact

scholarly journals Convergence results for two Kähler–Ricci flows

2014 ◽  
Vol 25 (09) ◽  
pp. 1450084
Author(s):  
Zhou Zhang
Keyword(s):  
Ricci Curvature ◽  
Einstein Metrics ◽  
Curvature Form ◽  
Volume Form ◽  
Ricci Flows ◽  
Convergence Results ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics

In this note, we provide some general discussion on the two main versions in the study of Kähler–Ricci flows over closed manifolds, aiming at smooth convergence to the corresponding Kähler–Einstein metrics with assumptions on the volume form and Ricci curvature form along the flow.

scholarly journals On the long time behaviour of the conical Kähler–Ricci flows

2016 ◽  
Vol 0 (0) ◽  
Author(s):  
Xiuxiong Chen ◽  
Yuanqi Wang
Keyword(s):  
Einstein Metrics ◽  
Maximal Regularity ◽  
Type Theorem ◽  
Conical Singularities ◽  
Ricci Flows ◽  
Long Time Behaviour ◽  
Long Time ◽  
First Chern Class ◽  
Kähler Einstein Metrics ◽  
Key Steps

Abstract We prove that the conical Kähler–Ricci flows introduced in [11] exist for all time {t\in[0,+\infty)} . These immortal flows possess maximal regularity in the conical category. As an application, we show if the twisted first Chern class {C_{1,\beta}} is negative or zero, the corresponding conical Kähler–Ricci flows converge to Kähler–Einstein metrics with conical singularities exponentially fast. To establish these results, one of our key steps is to prove a Liouville-type theorem for Kähler–Ricci flat metrics (which are defined over {\mathbb{C}^{n}} ) with conical singularities.

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