Reverse comparison theorems with upper integral Ricci curvature condition

2022 ◽  
Author(s):  
Hang Chen ◽  
Chaoqun Gao
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorems ◽  
Curvature Condition

scholarly journals Volume comparison of Bishop-Gromov type

1992 ◽  
Vol 45 (2) ◽  
pp. 241-248
Author(s):  
Sungyun Lee
Keyword(s):  
Riemannian Manifold ◽  
Ricci Curvature ◽  
Complete Riemannian Manifold ◽  
Comparison Theorems ◽  
Volume Comparison

Bishop-Gromov type comparison theorems for the volume of a tube about a sub-manifold of a complete Riemannian manifold whose Ricci curvature is bounded from below are proved. The Kähler analogue is also proved.

A note on a three-dimensional sphere theorem with integral curvature condition

2016 ◽  
Vol 18 (04) ◽  
pp. 1550070 ◽  
Author(s):  
Mijia Lai
Keyword(s):  
Scalar Curvature ◽  
Ricci Curvature ◽  
Space Form ◽  
Three Dimensional ◽  
Dimensional Sphere ◽  
Integral Curvature ◽  
Spherical Space ◽  
Sphere Theorem ◽  
Curvature Condition ◽  
Spherical Space Form

In this paper, we obtain a three-dimensional sphere theorem with integral curvature condition. On a closed three manifold [Formula: see text] with constant positive scalar curvature, if a certain combination of [Formula: see text] norm of the Ricci curvature and [Formula: see text] norm of the scalar curvature is positive, then [Formula: see text] is diffeomorphic to a spherical space form.

scholarly journals Comparison Geometry of Manifolds with Boundary under a Lower Weighted Ricci Curvature Bound

2018 ◽  
Vol 72 (1) ◽  
pp. 243-280
Author(s):  
Yohei Sakurai
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Suitable Condition ◽  
Manifolds With Boundary ◽  
Weighted Mean ◽  
Curvature Condition ◽  
Curvature Bound ◽  
Comparison Geometry ◽  
Weighted Mean Curvature ◽  
Weighted Ricci Curvature

AbstractWe study Riemannian manifolds with boundary under a lower weighted Ricci curvature bound. We consider a curvature condition in which the weighted Ricci curvature is bounded from below by the density function. Under the curvature condition and a suitable condition for the weighted mean curvature for the boundary, we obtain various comparison geometric results.

scholarly journals On interpolation and curvature via Wasserstein geodesics

2017 ◽  
Vol 10 (2) ◽  
pp. 125-167 ◽  
Author(s):  
Martin Kell
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Similar Condition ◽  
Interpolation Inequality ◽  
Metric Measure Spaces ◽  
Main Ingredient ◽  
Curvature Condition ◽  
Finsler Manifolds ◽  
Measure Spaces ◽  
Positively Curved

AbstractIn this article, a proof of the interpolation inequality along geodesics in p-Wasserstein spaces is given. This interpolation inequality was the main ingredient to prove the Borel–Brascamp–Lieb inequality for general Riemannian and Finsler manifolds and led Lott–Villani and Sturm to define an abstract Ricci curvature condition. Following their ideas, a similar condition can be defined and for positively curved spaces one can prove a Poincaré inequality. Using Gigli’s recently developed calculus on metric measure spaces, even a q-Laplacian comparison theorem holds on q-infinitesimal convex spaces. In the appendix, the theory of Orlicz–Wasserstein spaces is developed and necessary adjustments to prove the interpolation inequality along geodesics in those spaces are given.

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