A pointwise gradient bound for elliptic equations on compact manifolds with nonnegative Ricci curvature

2011 ◽  
Vol 30 (4) ◽  
pp. 1139-1144 ◽  
Author(s):  
Alberto Farina ◽  
◽  
Enrico Valdinoci ◽  
Keyword(s):  
Ricci Curvature ◽  
Elliptic Equations ◽  
Compact Manifolds ◽  
Nonnegative Ricci Curvature ◽  
Gradient Bound

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 Rigidity of volume-minimising hypersurfaces in Riemannian 5-manifolds

2018 ◽  
Vol 167 (02) ◽  
pp. 345-353
Author(s):  
ABRAÃO MENDES
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Positive Constant ◽  
Homotopy Class ◽  
Upper Bound ◽  
Rigidity Result ◽  
Nonnegative Ricci Curvature ◽  
Closed Hypersurface

AbstractIn this paper we generalise the main result of [4] for manifolds that are not necessarily Einstein. In fact, we obtain an upper bound for the volume of a locally volume-minimising closed hypersurface Σ of a Riemannian 5-manifold M with scalar curvature bounded from below by a positive constant in terms of the total traceless Ricci curvature of Σ. Furthermore, if Σ saturates the respective upper bound and M has nonnegative Ricci curvature, then Σ is isometric to 𝕊4 up to scaling and M splits in a neighbourhood of Σ. Also, we obtain a rigidity result for the Riemannian cover of M when Σ minimises the volume in its homotopy class and saturates the upper bound.

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