scholarly journals Sasaki-Einstein space T 1,1, transverse Kähler-Ricci flow and Sasaki-Ricci soliton

2019 ◽  
Vol 1194 ◽  
pp. 012110
Author(s):  
Mihai Visinescu
Keyword(s):  
Ricci Flow ◽  
Ricci Soliton ◽  
Einstein Space

Sasaki–Ricci flow on Sasaki–Einstein space T1,1 and deformations

2018 ◽  
Vol 33 (34) ◽  
pp. 1845014 ◽  
Author(s):  
Mihai Visinescu
Keyword(s):  
Ricci Flow ◽  
Vector Fields ◽  
Hamiltonian Function ◽  
Ricci Soliton ◽  
Einstein Space ◽  
Reeb Vector ◽  
Kähler Structure ◽  
Reeb Vector Field ◽  
Holomorphic Vector Fields ◽  
Kähler Potential

We study the transverse Kähler structure of the Sasaki–Einstein space [Formula: see text]. A set of local holomorphic coordinates is introduced and a Sasakian analogue of the Kähler potential is given. We investigate deformations of the Sasaki–Einstein structure preserving the Reeb vector field, but modifying the contact form. For this kind of deformations, we consider the Sasaki–Ricci flow which converges in a suitable sense to a Sasaki–Ricci soliton. Finally, it is described the constructions of Hamiltonian holomorphic vector fields and Hamiltonian function on the [Formula: see text] manifold.

scholarly journals BOUNDING SECTIONAL CURVATURE ALONG THE KÄHLER–RICCI FLOW

2009 ◽  
Vol 11 (06) ◽  
pp. 1067-1077 ◽  
Author(s):  
WEI-DONG RUAN ◽  
YUGUANG ZHANG ◽  
ZHENLEI ZHANG
Keyword(s):  
Sectional Curvature ◽  
Chern Class ◽  
Ricci Flow ◽  
Kähler Manifold ◽  
Ricci Soliton ◽  
Curvature Operator ◽  
Kahler Manifold ◽  
First Chern Class ◽  
Compact Kähler Manifold ◽  
Uniformly Bounded

If a normalized Kähler–Ricci flow g(t), t ∈ [0,∞), on a compact Kähler manifold M, dim ℂ M = n ≥ 3, with positive first Chern class satisfies g(t) ∈ 2πc1(M) and has curvature operator uniformly bounded in Ln-norm, the curvature operator will also be uniformly bounded along the flow. Consequently, the flow will converge along a subsequence to a Kähler–Ricci soliton.

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].

scholarly journals Homogeneous Ricci solitons

2015 ◽  
Vol 2015 (699) ◽  
Author(s):  
Michael Jablonski
Keyword(s):  
Lie Groups ◽  
Ricci Flow ◽  
Isometry Group ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Solvable Lie Groups ◽  
Compact Case ◽  
Group Of Isometries ◽  
Simply Connected ◽  
Special Case

AbstractIn this work, we study metrics which are both homogeneous and Ricci soliton. If there exists a transitive solvable group of isometries on a Ricci soliton, we show that it is isometric to a solvsoliton. Moreover, unless the manifold is flat, it is necessarily simply-connected and diffeomorphic to ℝIn the general case, we prove that homogeneous Ricci solitons must be semi-algebraic Ricci solitons in the sense that they evolve under the Ricci flow by dilation and pullback by automorphisms of the isometry group. In the special case that there exists a transitive semi-simple group of isometries on a Ricci soliton, we show that such a space is in fact Einstein. In the compact case, we produce new proof that Ricci solitons are necessarily Einstein.Lastly, we characterize solvable Lie groups which admit Ricci soliton metrics.

scholarly journals Sasaki–Ricci flow equation on five-dimensional Sasaki–Einstein space Yp,q

2020 ◽  
Vol 35 (14) ◽  
pp. 2050114
Author(s):  
Mihai Visinescu
Keyword(s):  
Ricci Flow ◽  
Contact Structure ◽  
Flow Equation ◽  
Einstein Space ◽  
Explicit Solutions

We analyze the transverse Kähler–Ricci flow equation on Sasaki-Einstein space [Formula: see text]. Explicit solutions are produced representing new five-dimensional Sasaki structures. Solutions which do not modify the transverse metric preserve the Sasaki–Einstein feature of the contact structure. If the transverse metric is altered, the deformed metrics remain Sasaki, but not Einstein.

scholarly journals Randers Ricci soliton homogeneous nilmanifolds

2021 ◽  
Vol 127 (1) ◽  
pp. 100-110
Author(s):  
Hamid Reza Salimi Moghaddam
Keyword(s):  
Vector Field ◽  
Lie Group ◽  
Ricci Flow ◽  
Ricci Soliton ◽  
Riemannian Metric ◽  
Flow Equation ◽  
Nilpotent Lie Group ◽  
Randers Metric ◽  
Left Invariant Riemannian Metric ◽  
Simply Connected

Let $F$ be a left-invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left-invariant Riemannian metric $\hat{\boldsymbol{a}}$ and a vector field $X$ which is $I_{\hat{\boldsymbol{a}}}(M)$-invariant. We show that if the Ricci flow equation has a unique solution then, $(N,F)$ is a Ricci soliton if and only if $(N,F)$ is a semialgebraic Ricci soliton.

scholarly journals SOME RESULTS ON ∗−RICCI FLOW

2021 ◽  
pp. 1305
Author(s):  
Dipankar Debnath ◽  
Nirabhra Basu
Keyword(s):  
Ricci Flow ◽  
Ricci Soliton ◽  
Geometric Curvature ◽  
Self Similar ◽  
Curvature Tensors

In this paper we have introduced the notion of ∗-Ricci flow and shown that ∗-Ricci soliton which was introduced by Kaimakamis and Panagiotidou in 2014 which is a self similar soliton of the ∗-Ricci flow. We have also find the deformation of geometric curvature tensors under ∗-Ricci flow. In the last two section of the paper, we have found the F-functional and ω-functional for ∗-Ricci flow respectively.

scholarly journals Kähler–Ricci flow on homogeneous toric bundles

2020 ◽  
Vol 31 (03) ◽  
pp. 2050022
Author(s):  
Hong Huang
Keyword(s):  
Lie Group ◽  
Ricci Flow ◽  
Ricci Soliton ◽  
Torus Action ◽  
Toric Manifold ◽  
Semisimple Lie Group ◽  
Dimension Formula ◽  
Complex Dimension ◽  
Algebraic Torus ◽  
Kähler Form

Assume that [Formula: see text] is a homogeneous toric bundle of the form [Formula: see text] and is Fano, where [Formula: see text] is a compact semisimple Lie group with complexification [Formula: see text], [Formula: see text] a parabolic subgroup of [Formula: see text], [Formula: see text] is a surjective homomorphism from [Formula: see text] to the algebraic torus [Formula: see text], and [Formula: see text] is a compact toric manifold of complex dimension [Formula: see text]. In this note, we show that the normalized Kähler–Ricci flow on [Formula: see text] with a [Formula: see text]-invariant initial Kähler form in [Formula: see text] converges, modulo the algebraic torus action, to a Kähler–Ricci soliton. This extends a previous work of Zhu. As a consequence, we recover a result of Podestà–Spiro.

scholarly journals The Fundamental Groups of m-Quasi-Einstein Manifolds

ISRN Geometry ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-4 ◽  
Author(s):  
Hee Kwon Lee
Keyword(s):  
Fundamental Group ◽  
Positive Constant ◽  
Ricci Flow ◽  
Ricci Tensor ◽  
Flow Theory ◽  
Ricci Soliton ◽  
Fundamental Groups ◽  
Einstein Manifolds ◽  
Constant Multiple ◽  
Metric Tensor

In Ricci flow theory, the topology of Ricci soliton is important. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of gradient shrinking Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of m-quasi-Einstein with a positive constant multiple.

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