scholarly journals Injectivity radius bound of Ricci flow with positive Ricci curvature and applications

2013 ◽  
Vol 8 (5) ◽  
pp. 1129-1137
Author(s):  
Li Ma ◽  
Anqiang Zhu
Keyword(s):  
Ricci Curvature ◽  
Ricci Flow ◽  
Injectivity Radius ◽  
Positive Ricci Curvature

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 Density Problems for second order Sobolev Spaces and Cut-off Functions on Manifolds With Unbounded Geometry

2019 ◽  
Author(s):  
Debora Impera ◽  
Michele Rimoldi ◽  
Giona Veronelli
Keyword(s):  
Maximum Principle ◽  
Ricci Curvature ◽  
Sobolev Spaces ◽  
Upper Bound ◽  
Ricci Tensor ◽  
Injectivity Radius ◽  
Quadratic Growth ◽  
Riemann Curvature ◽  
Main Application ◽  
Compactly Supported Functions

Abstract We consider complete non-compact manifolds with either a sub-quadratic growth of the norm of the Riemann curvature, or a sub-quadratic growth of both the norm of the Ricci curvature and the squared inverse of the injectivity radius. We show the existence on such a manifold of a distance-like function with bounded gradient and mild growth of the Hessian. As a main application, we prove that smooth compactly supported functions are dense in $W^{2,p}$. The result is improved for $p=2$ avoiding both the upper bound on the Ricci tensor, and the injectivity radius assumption. As further applications we prove new disturbed Sobolev and Calderón–Zygmund inequalities on manifolds with possibly unbounded curvature and highlight consequences about the validity of the full Omori–Yau maximum principle for the Hessian.

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