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

Holomorphic Functions on a Class of Kähler Manifolds With Nonnegative Curvature, I

2020 ◽  
Author(s):  
Bo Yang
Keyword(s):  
Asymptotic Behavior ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Polynomial Growth ◽  
Holomorphic Functions ◽  
Nonnegative Curvature ◽  
Kähler Manifolds ◽  
Kahler Manifolds ◽  
Nonnegative Ricci Curvature

Abstract In this paper, we consider holomorphic functions of polynomial growth on complete Kähler manifolds with nonnegative curvature. We explain how their growth orders are related to the asymptotic behavior of Kähler–Ricci flow. The main result is to determine minimal orders of holomorphic functions on gradient Kähler–Ricci expanding solitons with nonnegative Ricci curvature.

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.

scholarly journals Almost Euclidean Isoperimetric Inequalities in Spaces Satisfying Local Ricci Curvature Lower Bounds

2018 ◽  
Vol 2020 (5) ◽  
pp. 1481-1510 ◽  
Author(s):  
Fabio Cavalletti ◽  
Andrea Mondino
Keyword(s):  
Riemannian Manifold ◽  
Isoperimetric Inequality ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Ricci Flow ◽  
General Framework ◽  
Isoperimetric Inequalities ◽  
Optimal Transportation ◽  
Maximal Volume ◽  
Bounded Below

Abstract Motivated by Perelman’s Pseudo-Locality Theorem for the Ricci flow, we prove that if a Riemannian manifold has Ricci curvature bounded below in a metric ball which moreover has almost maximal volume, then in a smaller ball (in a quantified sense) it holds an almost euclidean isoperimetric inequality. The result is actually established in the more general framework of non-smooth spaces satisfying local Ricci curvature lower bounds in a synthetic sense via optimal transportation.

scholarly journals On Scalar and Ricci Curvatures

2021 ◽  
Author(s):  
Gerard Besson ◽  
◽  
Sylvestre Gallot ◽  
◽  
◽  
...  
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Lower Bounds ◽  
Riemannian Metric ◽  
Metric Measure Spaces ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces ◽  
Volume Entropy ◽  
Bounded Below

The purpose of this report is to acknowledge the influence of M. Gromov's vision of geometry on our own works. It is two-fold: in the first part we aim at describing some results, in dimension 3, around the question: which open 3-manifolds carry a complete Riemannian metric of positive or non negative scalar curvature? In the second part we look for weak forms of the notion of ''lower bounds of the Ricci curvature'' on non necessarily smooth metric measure spaces. We describe recent results some of which are already posted in [arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt to give a new synthetic version of Ricci curvature bounded below using Bishop-Gromov's inequality.

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.

Rigidity in dimension four of area-minimising Einstein manifolds

2015 ◽  
Vol 158 (2) ◽  
pp. 355-363 ◽  
Author(s):  
A. BARROS ◽  
C. CRUZ ◽  
R. BATISTA ◽  
P. SOUSA
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Einstein Manifold ◽  
Sharp Inequality ◽  
Positive Scalar Curvature ◽  
Suitable Choice ◽  
Dimensional Manifold ◽  
Einstein Manifolds ◽  
Standard Sphere ◽  
Nonnegative Ricci Curvature

AbstractThe aim of this paper is to prove a sharp inequality for the area of a four dimensional compact Einstein manifold (Σ,gΣ) embedded into a complete five dimensional manifold (M5,g) with positive scalar curvatureRand nonnegative Ricci curvature. Under a suitable choice, we have$area(\Sigma)^{\frac{1}{2}}\inf_{M}R \leq 8\sqrt{6}\pi$. Moreover, occurring equality we deduce that (Σ,gΣ) is isometric to a standard sphere ($\mathbb{S}$4,gcan) and in a neighbourhood of Σ, (M5,g) splits as ((-ϵ, ϵ) ×$\mathbb{S}$4,dt2+gcan) and the Riemannian covering of (M5,g) is isometric to$\Bbb{R}$×$\mathbb{S}$4.

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