scholarly journals Lower volume growth and total σk-scalar curvature estimates

2015 ◽  
Vol 42 ◽  
pp. 104-114
Author(s):  
Márcio Batista ◽  
Heudson Mirandola
Keyword(s):  
Scalar Curvature ◽  
Volume Growth ◽  
Curvature Estimates ◽  
Lower Volume

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 Local curvature estimates for the Laplacian flow

2021 ◽  
Vol 60 (1) ◽  
Author(s):  
Yi Li
Keyword(s):  
Scalar Curvature ◽  
Elementary Proof ◽  
Time Derivative ◽  
Main Ingredient ◽  
Local Curvature ◽  
Curvature Estimates ◽  
The Difference ◽  
Complete Manifolds ◽  
Laplacian Flow ◽  
Funct Anal

AbstractIn this paper we give local curvature estimates for the Laplacian flow on closed $$G_{2}$$ G 2 -structures under the condition that the Ricci curvature is bounded along the flow. The main ingredient consists of the idea of Kotschwar et al. (J Funct Anal 271(9):2604–2630, 2016) who gave local curvature estimates for the Ricci flow on complete manifolds and then provided a new elementary proof of Sesum’s result (Sesum in Am J Math 127(6):1315–1324, 2005), and the particular structure of the Laplacian flow on closed $$G_{2}$$ G 2 -structures. As an immediate consequence, this estimates give a new proof of Lotay and Wei’s (Geom Funct Anal 27(1):165–233, 2017) result which is an analogue of Sesum’s theorem. The second result is about an interesting evolution equation for the scalar curvature of the Laplacian flow of closed $$G_{2}$$ G 2 -structures. Roughly speaking, we can prove that the time derivative of the scalar curvature $$R_{g(t)}$$ R g ( t ) is equal to the Laplacian of $$R_{g(t)}$$ R g ( t ) , plus an extra term which can be written as the difference of two nonnegative quantities.

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