scholarly journals Bergman kernel along the Kähler–Ricci flow and Tian’s conjecture

2016 ◽  
Vol 2016 (717) ◽  
Author(s):  
Wenshuai Jiang
Keyword(s):  
Recent Work ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Bergman Kernel ◽  
Fano Manifolds ◽  
Bergman Kernels ◽  
Complex Dimension ◽  
Short Time

AbstractIn this paper, we study the behavior of Bergman kernels along the Kähler–Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler–Ricci flow for short time under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture of Tian for Fano manifolds of complex dimension at most 3.

scholarly journals POSITIVITY OF RICCI CURVATURE UNDER THE KÄHLER–RICCI FLOW

2006 ◽  
Vol 08 (01) ◽  
pp. 123-133 ◽  
Author(s):  
DAN KNOPF
Keyword(s):  
Ricci Curvature ◽  
Ricci Flow ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Bounded Curvature ◽  
Complex Dimension

In each complex dimension n ≥ 2, we construct complete Kähler manifolds of bounded curvature and non-negative Ricci curvature whose Kähler–Ricci evolutions immediately acquire Ricci curvature of mixed sign.

scholarly journals Ricci flow on manifolds with boundary with arbitrary initial metric

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tsz-Kiu Aaron Chow
Keyword(s):  
Ricci Flow ◽  
Positive Curvature ◽  
Curvature Operator ◽  
Manifolds With Boundary ◽  
Convex Boundary ◽  
Uniqueness Of The Solution ◽  
Positive Time ◽  
Natural Boundary Conditions ◽  
Short Time ◽  
Time Existence

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

scholarly journals Lower bounds on Ricci flow invariant curvatures and geometric applications

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Thomas Richard
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Nonnegative Ricci Curvature ◽  
Invariant Cones

AbstractWe consider Ricci flow invariant cones 𝒞 in the space of curvature operators lying between the cones “nonnegative Ricci curvature” and “nonnegative curvature operator”. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies

scholarly journals On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold

2012 ◽  
Vol 148 (6) ◽  
pp. 1985-2003 ◽  
Author(s):  
Chi Li
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Limit Behavior ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Continuity Method ◽  
Type Singularity ◽  
Moment Polytope ◽  
The Moment ◽  
Monge Ampere

AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

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.

scholarly journals RICCI LOWER BOUND FOR KÄHLER–RICCI FLOW

2014 ◽  
Vol 16 (02) ◽  
pp. 1350053 ◽  
Author(s):  
ZHOU ZHANG
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Finite Time ◽  
Ricci Flows ◽  
Time Singularities

We provide general discussion on the lower bound of Ricci curvature along Kähler–Ricci flows over closed manifolds. The main result is the non-existence of Ricci lower bound for flows with finite time singularities and non-collapsed global volume. As an application, we give examples showing that positivity of Ricci curvature would not be preserved by Ricci flow in general.

scholarly journals Interstellar Formaldehyde

1971 ◽  
Vol 2 ◽  
pp. 394-401 ◽  
Author(s):  
Patrick Palmer
Keyword(s):  
Energy Level ◽  
Interstellar Medium ◽  
Recent Work ◽  
Large Scale ◽  
Galactic Structure ◽  
Isotopic Abundance ◽  
Microwave Background ◽  
Initial Discovery ◽  
Short Time

Formaldehyde (H2CO) was found to be present in the interstellar medium less than 1½ yr ago. This discovery raised a number of new astrophysical problems specific to H2CO such as how is the molecule formed, how is it destroyed, and how do the anomalous energy level populations leading to absorption of the isotropic microwave background arise. In addition, H2CO has provided a new means of studying several problems of long standing interest in astronomy. These include large scale galactic structure, distribution and motions of local gas and dust, and isotopic abundance ratios. A surprisingly large amount of work has been done on these problems in the rather short time since the initial discovery. I will confine myself to a brief summary of the observational aspects, omitting many of the topics that will be discussed in more detail by others later today, and then describe in detail some of our more recent work.

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