scholarly journals Convergence of the Kähler–Ricci Flow on a Kähler–Einstein Fano Manifold

2013 ◽  
pp. 299-333
Author(s):  
Vincent Guedj
Keyword(s):  
Ricci Flow ◽  
Fano Manifold

scholarly journals On the Kähler–Ricci flow near a Kähler–Einstein metric

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Song Sun ◽  
Yuanqi Wang
Keyword(s):  
Critical Point ◽  
Ricci Flow ◽  
Complex Structure ◽  
Type Inequality ◽  
Gradient Flow ◽  
Einstein Metric ◽  
Fano Manifold ◽  
Kähler Metric ◽  
Canonical Class ◽  
Kahler Metric

AbstractOn a Fano manifold, we prove that the Kähler–Ricci flow starting from a Kähler metric in the anti-canonical class which is sufficiently close to a Kähler–Einstein metric must converge in a polynomial rate to a Kähler–Einstein metric. The convergence cannot happen in general if we study the flow on the level of Kähler potentials. Instead we exploit the interpretation of the Ricci flow as the gradient flow of Perelman's μ functional. This involves modifying the Ricci flow by a canonical family of gauges. In particular, the complex structure of the limit could be different in general. The main technical ingredient is a Lojasiewicz type inequality for Perelman's μ functional near a critical point.

scholarly journals Perelman’s entropy and Kähler-Ricci flow on a Fano manifold

2013 ◽  
Vol 365 (12) ◽  
pp. 6669-6695 ◽  
Author(s):  
Gang Tian ◽  
Shijin Zhang ◽  
Zhenlei Zhang ◽  
Xiaohua Zhu
Keyword(s):  
Ricci Flow ◽  
Fano Manifold

scholarly journals The Soliton-Ricci Flow with variable volume forms

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Nefton Pali
Keyword(s):  
Ricci Flow ◽  
Riemannian Metrics ◽  
Gradient Flow ◽  
Ricci Soliton ◽  
Riemannian Structure ◽  
Fano Manifold ◽  
Finite Dimensional Reduction ◽  
Positive Volume ◽  
Formal Limit ◽  
Volume Forms

AbstractWe introduce a flow of Riemannian metrics and positive volume forms over compact oriented manifolds whose formal limit is a shrinking Ricci soliton. The case of a fixed volume form has been considered in our previouswork.We still call this new flow, the Soliton-Ricci flow. It corresponds to a forward Ricci type flow up to a gauge transformation. This gauge is generated by the gradient of the density of the volumes. The new Soliton-Ricci flow exist for all times. It represents the gradient flow of Perelman’s W functional with respect to a pseudo-Riemannian structure over the space of metrics and normalized positive volume forms. We obtain an expression of the Hessian of the W functional with respect to such structure. Our expression shows the elliptic nature of this operator in the orthogonal directions to the orbits obtained by the action of the group of diffeomorphism. In the case that initial data is Kähler, the Soliton-Ricci flow over a Fano manifold preserves the Kähler condition and the symplectic form. Over a Fano manifold, the space of tamed complex structures embeds naturally, via the Chern-Ricci map, into the space of metrics and normalized positive volume forms. Over such space the pseudo-Riemannian structure restricts to a Riemannian one. We perform a study of the sign of the restriction of the Hessian of the W functional over such space. This allows us to obtain a finite dimensional reduction of the stability problem for Kähler-Ricci solitons. This reduction represents the solution of this well known problem. A less precise and less geometric version of this result has been obtained recently by the author in [28].

Stability of Kähler-Ricci flow on a Fano manifold

2012 ◽  
Vol 356 (4) ◽  
pp. 1425-1454 ◽  
Author(s):  
Xiaohua Zhu
Keyword(s):  
Ricci Flow ◽  
Fano Manifold

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.

scholarly journals Sliding Mode Control and Geometrization Conjecture in Seismic Response

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 353
Author(s):  
Ligia Munteanu ◽  
Dan Dumitriu ◽  
Cornel Brisan ◽  
Mircea Bara ◽  
Veturia Chiroiu ◽  
...  
Keyword(s):  
Sliding Mode Control ◽  
Ricci Flow ◽  
Lyapunov Functions ◽  
Seismic Waves ◽  
Sliding Mode ◽  
Flow Process ◽  
Mode Control ◽  
Finite Period ◽  
The Stability ◽  
Story Building

The purpose of this paper is to study the sliding mode control as a Ricci flow process in the context of a three-story building structure subjected to seismic waves. The stability conditions result from two Lyapunov functions, the first associated with slipping in a finite period of time and the second with convergence of trajectories to the desired state. Simulation results show that the Ricci flow control leads to minimization of the displacements of the floors.

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