Mini-Workshop: Einstein Metrics, Ricci Solitons and Ricci Flow under Symmetry Assumptions

2014 ◽  
Vol 11 (4) ◽  
pp. 2529-2568
Author(s):  
Christoph Böhm ◽  
Jorge Lauret ◽  
McKenzie Wang
Keyword(s):  
Ricci Flow ◽  
Einstein Metrics ◽  
Ricci Solitons

Gap theorems for compact almost Ricci-harmonic solitons

2019 ◽  
Vol 30 (08) ◽  
pp. 1950040 ◽  
Author(s):  
Abimbola Abolarinwa
Keyword(s):  
Einstein Metrics ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Ricci Solitons ◽  
Compact Domain ◽  
Necessary And Sufficient ◽  
Gap Theorems

Almost Ricci-harmonic solitons are generalization of Ricci-harmonic solitons, almost Ricci solitons and harmonic-Einstein metrics. The main focus of this paper is to establish necessary and sufficient conditions for a gradient shrinking almost Ricci-harmonic soliton on a compact domain to be almost harmonic-Einstein.

THE SASAKI–RICCI FLOW

2010 ◽  
Vol 21 (07) ◽  
pp. 951-969 ◽  
Author(s):  
KNUT SMOCZYK ◽  
GUOFANG WANG ◽  
YONGBING ZHANG
Keyword(s):  
Ricci Flow ◽  
Einstein Metrics ◽  
Positive Case ◽  
Einstein Metric ◽  
Well Posedness ◽  
Long Time Existence ◽  
Null Case ◽  
Long Time ◽  
Time Existence

In this paper, we introduce the Sasaki–Ricci flow to study the existence of η-Einstein metrics. In the positive case any η-Einstein metric can be homothetically transformed to a Sasaki–Einstein metric. Hence it is an odd-dimensional counterpart of the Kähler–Ricci flow. We prove its well-posedness and long-time existence. In the negative or null case the flow converges to the unique η-Einstein metric. In the positive case the convergence remains in general open. The paper can be viewed as an odd-dimensional counterpart of Cao's results on the Kähler–Ricci flow.

scholarly journals Kähler–Einstein metrics and the Kähler–Ricci flow on log Fano varieties

2019 ◽  
Vol 2019 (751) ◽  
pp. 27-89 ◽  
Author(s):  
Robert J. Berman ◽  
Sebastien Boucksom ◽  
Philippe Eyssidieux ◽  
Vincent Guedj ◽  
Ahmed Zeriahi
Keyword(s):  
Ricci Flow ◽  
Existence And Uniqueness ◽  
Additional Condition ◽  
Einstein Metrics ◽  
Discrete Version ◽  
Fano Varieties ◽  
Fano Manifolds ◽  
Smooth Convergence ◽  
Kähler Einstein Metrics ◽  
Special Case

AbstractWe prove the existence and uniqueness of Kähler–Einstein metrics on {{\mathbb{Q}}}-Fano varieties with log terminal singularities (and more generally on log Fano pairs) whose Mabuchi functional is proper. We study analogues of the works of Perelman on the convergence of the normalized Kähler–Ricci flow, and of Keller, Rubinstein on its discrete version, Ricci iteration. In the special case of (non-singular) Fano manifolds, our results on Ricci iteration yield smooth convergence without any additional condition, improving on previous results. Our result for the Kähler–Ricci flow provides weak convergence independently of Perelman’s celebrated estimates.

RICCI FLOW EQUATION ON C-REDUCIBLE METRICS

2011 ◽  
Vol 08 (04) ◽  
pp. 773-781 ◽  
Author(s):  
NASRIN SADEGHZADEH ◽  
ASSADOLLAH RAZAVI
Keyword(s):  
Fixed Point ◽  
Ricci Flow ◽  
Einstein Metrics ◽  
Flow Equation ◽  
Finsler Metrics

This paper focuses on the study of a deformation of Finsler metrics satisfying Ricci flow equation. We will prove that every deformation of Randers (Kropina)-metrics satisfying Ricci flow equation is Einstein, we will also show that a deformation of Einstein metrics with initial Randers (Kropina)-metrics remains Randers (Kropina). In other words the deformation of Randers (or Kropina)-metrics is exactly the fixed point of (un-normal and normal) Ricci flow equation.

scholarly journals Non-Kähler Ricci flow singularities modeled on Kähler–Ricci solitons

2019 ◽  
Vol 15 (2) ◽  
pp. 749-784 ◽  
Author(s):  
James Isenberg ◽  
Dan Knopf ◽  
Nataša Šešum
Keyword(s):  
Ricci Flow ◽  
Ricci Solitons

A CHARACTERIZATION OF NONCOMPACT KOISO-TYPE SOLITONS

2012 ◽  
Vol 23 (05) ◽  
pp. 1250054 ◽  
Author(s):  
BO YANG
Keyword(s):  
Einstein Metrics ◽  
Line Bundles ◽  
Ricci Solitons ◽  
Academic Press ◽  
Einstein Manifolds ◽  
Pure Mathematics ◽  
Rotationally Symmetric ◽  
Kähler Einstein Metrics ◽  
Holomorphic Line Bundles

We construct complete gradient Kähler–Ricci solitons of various types on the total spaces of certain holomorphic line bundles over compact Kähler–Einstein manifolds with positive scalar curvature. Those are noncompact analogues of the compact examples found by Koiso [On rotationally symmetric Hamilton's equations for Kähler–Einstein metrics, in Recent Topics in Differential and Analytic Geometry, Advanced Studies in Pure Mathematics, Vol. 18-I (Academic Press, Boston, MA, 1990), pp. 327–337]. Our examples can be viewed a generalization of previous examples by Cao [Existense of gradient Kähler–Ricci solitons, in Elliptic and Parabolic Methods in Geometry (Minneapolis, MN, 1994), pp. 1–16], Chave and Valent [On a class of compact and non-compact quasi-Einstein metrics and their renormalizability properties, Nuclear Phys. B 478 (1996) 758–778], Pedersen, Tønnesen-Friedman, and Valent [Quasi-Einstein Kähler metrics, Lett. Math. Phys. 50(3) (1999) 229–241], and Feldman, Ilmanen and Knopf [Rotationally symmetric shrinking and expanding gradient Kähler–Ricci solitons, J. Differential Geom. 65 (2003) 169–209]. We also prove a uniformization result on complete steady gradient Kähler–Ricci solitons with non-negative Ricci curvature under additional assumptions.

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