Three circle theorems for eigenfunctions on complete shrinking gradient Ricci solitons with constant scalar curvature

2018 ◽  
Vol 464 (2) ◽  
pp. 1243-1259
Author(s):  
Jianyu Ou
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons

scholarly journals Ricci solitons and gradient Ricci solitons in three-dimensional trans-Sasakian manifolds

Filomat ◽  
2012 ◽  
Vol 26 (2) ◽  
pp. 363-370 ◽  
Author(s):  
Mine Turan ◽  
Chand De ◽  
Ahmet Yildiz
Keyword(s):  
Scalar Curvature ◽  
Constant Scalar Curvature ◽  
Sasakian Manifold ◽  
Ricci Soliton ◽  
Ricci Solitons ◽  
Sasakian Manifolds ◽  
Kenmotsu Manifold ◽  
Constant Multiple ◽  
3 Dimensional ◽  
Gradient Ricci Solitons

The object of the present paper is to study 3-dimensional trans-Sasakian manifolds admitting Ricci solitons and gradient Ricci solitons. We prove that if (1,V, ?) is a Ricci soliton where V is collinear with the characteristic vector field ?, then V is a constant multiple of ? and the manifold is of constant scalar curvature provided ?, ? =constant. Next we prove that in a 3-dimensional trans-Sasakian manifold with constant scalar curvature if 1 is a gradient Ricci soliton, then the manifold is either a ?-Kenmotsu manifold or an Einstein manifold. As a consequence of this result we obtain several corollaries.

scholarly journals Four-dimensional complete gradient shrinking Ricci solitons

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Huai-Dong Cao ◽  
Ernani Ribeiro Jr ◽  
Detang Zhou
Keyword(s):  
Scalar Curvature ◽  
Potential Function ◽  
Weyl Tensor ◽  
Ricci Soliton ◽  
The Self ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons ◽  
Curvature Estimates ◽  
Finite Quotient

Abstract In this article, we study four-dimensional complete gradient shrinking Ricci solitons. We prove that a four-dimensional complete gradient shrinking Ricci soliton satisfying a pointwise condition involving either the self-dual or anti-self-dual part of the Weyl tensor is either Einstein, or a finite quotient of either the Gaussian shrinking soliton ℝ 4 {\mathbb{R}^{4}} , or 𝕊 3 × ℝ {\mathbb{S}^{3}\times\mathbb{R}} , or 𝕊 2 × ℝ 2 . {\mathbb{S}^{2}\times\mathbb{R}^{2}.} In addition, we provide some curvature estimates for four-dimensional complete gradient Ricci solitons assuming that its scalar curvature is suitable bounded by the potential function.

scholarly journals Estimates on the non-compact expanding gradient Ricci solitons

2013 ◽  
Vol 21 (3) ◽  
pp. 95-102
Author(s):  
Xiang Gao ◽  
Qiaofang Xing ◽  
Rongrong Cao
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Linear Growth ◽  
Ricci Soliton ◽  
Potential Functions ◽  
Ricci Solitons ◽  
Positive Ricci Curvature ◽  
Gradient Ricci Soliton ◽  
Gradient Ricci Solitons ◽  
Normal Geodesic

Abstract In this paper, we deal with the complete non-compact expanding gradient Ricci soliton (Mn,g) with positive Ricci curvature. On the condition that the Ricci curvature is positive and the scalar curvature approaches 0 towards infinity, we derive a useful estimate on the growth of potential functions. Based on this and under the same assumptions, we prove that ∫t0 Rc (γ'(s) , γ' (s))ds and ∫t0 Rc (γ' (,s). v)ds at least have linear growth, where 7(5) is a minimal normal geodesic emanating from the point where R obtains its maximum. Furthermore, some other results on the Ricci curvature are also obtained.

scholarly journals Generalizations of Kähler-Ricci solitons on projective bundles

2011 ◽  
Vol 108 (2) ◽  
pp. 161 ◽  
Author(s):  
Gideon Maschler ◽  
Christina W. Tønnesen-Friedman
Keyword(s):  
Scalar Curvature ◽  
Existence Theorem ◽  
Einstein Metrics ◽  
Constant Scalar Curvature ◽  
Ricci Solitons ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Projective Bundles ◽  
Affine Combination ◽  
Constant Scalar Curvature Metrics

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and Tønnesen-Friedman), arising from a base with a local Kähler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein Kähler metrics (as defined by D. Guan) in all "sufficiently small" admissible Kähler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some Kähler classes while not in others. We also prove an analogous existence theorem for an additional metric type, defined by the requirement that the scalar curvature is an affine combination of a Killing potential and its Laplacian.

scholarly journals Some results on Ricci-Bourguignon solitons and almost solitons

2020 ◽  
pp. 1-14
Author(s):  
Shubham Dwivedi
Keyword(s):  
Vector Field ◽  
Scalar Curvature ◽  
Integral Formula ◽  
Constant Scalar Curvature ◽  
Ricci Solitons ◽  
Euclidean Sphere ◽  
Integral Formulas

Abstract We prove some results for the solitons of the Ricci–Bourguignon flow, generalizing the corresponding results for Ricci solitons. Taking motivation from Ricci almost solitons, we then introduce the notion of Ricci–Bourguignon almost solitons and prove some results about them that generalize previous results for Ricci almost solitons. We also derive integral formulas for compact gradient Ricci–Bourguignon solitons and compact gradient Ricci–Bourguignon almost solitons. Finally, using the integral formula, we show that a compact gradient Ricci–Bourguignon almost soliton is isometric to a Euclidean sphere if it has constant scalar curvature or its associated vector field is conformal.

scholarly journals Geometry of shrinking Ricci solitons

2015 ◽  
Vol 151 (12) ◽  
pp. 2273-2300 ◽  
Author(s):  
Ovidiu Munteanu ◽  
Jiaping Wang
Keyword(s):  
Fixed Point ◽  
Lower Bound ◽  
Scalar Curvature ◽  
Distance Function ◽  
Upper Bound ◽  
Arbitrary Dimension ◽  
Injectivity Radius ◽  
Curvature Operator ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons

The main purpose of this paper is to investigate the curvature behavior of four-dimensional shrinking gradient Ricci solitons. For such a soliton $M$ with bounded scalar curvature $S$, it is shown that the curvature operator $\text{Rm}$ of $M$ satisfies the estimate $|\text{Rm}|\leqslant cS$ for some constant $c$. Moreover, the curvature operator $\text{Rm}$ is asymptotically nonnegative at infinity and admits a lower bound $\text{Rm}\geqslant -c(\ln (r+1))^{-1/4}$, where $r$ is the distance function to a fixed point in $M$. As an application, we prove that if the scalar curvature converges to zero at infinity, then the soliton must be asymptotically conical. As a separate issue, a diameter upper bound for compact shrinking gradient Ricci solitons of arbitrary dimension is derived in terms of the injectivity radius.

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