scholarly journals Correction to: Einstein four-manifolds of three-nonnegative curvature operator

2020 ◽  
Author(s):  
Peng Wu
Keyword(s):  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Four Manifolds

scholarly journals Lower bounds on Ricci flow invariant curvatures and geometric applications

2015 ◽  
Vol 2015 (703) ◽  
Author(s):  
Thomas Richard
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Nonnegative Ricci Curvature ◽  
Invariant Cones

AbstractWe consider Ricci flow invariant cones 𝒞 in the space of curvature operators lying between the cones “nonnegative Ricci curvature” and “nonnegative curvature operator”. Assuming some mild control on the scalar curvature of the Ricci flow, we show that if a solution to the Ricci flow has its curvature operator which satisfies

scholarly journals On the equivalence between noncollapsing and bounded entropy for ancient solutions to the Ricci flow

2020 ◽  
Vol 2020 (762) ◽  
pp. 35-51
Author(s):  
Yongjia Zhang
Keyword(s):  
Ricci Flow ◽  
Nonnegative Curvature ◽  
Curvature Operator ◽  
Type I ◽  
Curvature Bound ◽  
Reduced Volume ◽  
Ancient Solutions ◽  
Asymptotic Entropy ◽  
Uniformly Bounded

AbstractAs a continuation of a previous paper, we prove Perelman’s assertion, that is, for ancient solutions to the Ricci flow with bounded nonnegative curvature operator, uniformly bounded entropy is equivalent to κ-noncollapsing on all scales. We also establish an equality between the asymptotic entropy and the asymptotic reduced volume, which is a result similar to a paper by Xu (2017), where he assumes the Type I curvature bound.

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