On Shrinking Gradient Ricci Solitons with Positive Ricci Curvature and Small Weyl Tensor

2019 ◽  
Vol 39 (5) ◽  
pp. 1235-1239
Author(s):  
Zhuhong Zhang ◽  
Chih-Wei Chen
Keyword(s):  
Ricci Curvature ◽  
Weyl Tensor ◽  
Ricci Solitons ◽  
Positive Ricci Curvature ◽  
Gradient Ricci Solitons

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 The Weyl tensor of gradient Ricci solitons

2016 ◽  
Vol 20 (1) ◽  
pp. 389-436 ◽  
Author(s):  
Xiaodong Cao ◽  
Hung Tran
Keyword(s):  
Weyl Tensor ◽  
Ricci Solitons ◽  
Gradient Ricci Solitons

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 ON LOCALLY CONFORMALLY FLAT GRADIENT SHRINKING RICCI SOLITONS

2011 ◽  
Vol 13 (02) ◽  
pp. 269-282 ◽  
Author(s):  
XIAODONG CAO ◽  
BIAO WANG ◽  
ZHOU ZHANG
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Exponential Growth ◽  
Integral Identity ◽  
Ricci Solitons ◽  
Riemannian Curvature ◽  
Conformally Flat ◽  
Gradient Ricci Solitons ◽  
Riemannian Curvature Tensor ◽  
Locally Conformally Flat

In this paper, we first apply an integral identity on Ricci solitons to prove that closed locally conformally flat gradient Ricci solitons are of constant sectional curvature. We then generalize this integral identity to complete noncompact gradient shrinking Ricci solitons, under the conditions that the Ricci curvature is bounded from below and the Riemannian curvature tensor has at most exponential growth. As a consequence, we classify complete locally conformally flat gradient shrinking Ricci solitons with Ricci curvature bounded from below.

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