scholarly journals Improved Relative Volume Comparison for Integral Ricci Curvature and Applications to Volume Entropy

2021 ◽  
Vol 32 (1) ◽  
Author(s):  
Lina Chen ◽  
Guofang Wei
Keyword(s):  
Ricci Curvature ◽  
Relative Volume ◽  
Volume Comparison ◽  
Volume Entropy

Ricci curvature decay and upper bound of total mass

2019 ◽  
Vol 16 (12) ◽  
pp. 1950189
Author(s):  
Seong-Hun Paeng
Keyword(s):  
Ricci Curvature ◽  
Upper Bound ◽  
Total Mass ◽  
Relative Volume ◽  
Volume Comparison ◽  
Integral Norm ◽  
Weighted Integral ◽  
Curvature Decay

We obtain a positive upper bound of total mass from a relative volume comparison by a weighted integral norm of Ricci curvature.

ON INTEGRAL RICCI CURVATURE AND TOPOLOGY OF FINSLER MANIFOLDS

2012 ◽  
Vol 23 (11) ◽  
pp. 1250111 ◽  
Author(s):  
B. Y. WU
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Relative Volume ◽  
Volume Form ◽  
Finsler Manifolds ◽  
Volume Comparison ◽  
Minimal Volume ◽  
Curvature Bound ◽  
And Topology

We establish a relative volume comparison theorem for minimal volume form of Finsler manifolds under integral Ricci curvature bound. As its applications, we obtain some results on integral Ricci curvature and topology of Finsler manifolds. These results generalize the corresponding properties with pointwise Ricci curvature bound in the literatures.

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.

scholarly journals HARMONIC HADAMARD MANIFOLDS OF PRESCRIBED RICCI CURVATURE AND VOLUME ENTROPY

2016 ◽  
Vol 70 (2) ◽  
pp. 267-280
Author(s):  
Mitsuhiro ITOH ◽  
Sinhwi KIM ◽  
Jeonghyeong PARK ◽  
Hiroyasu SATOH
Keyword(s):  
Ricci Curvature ◽  
Hadamard Manifolds ◽  
Volume Entropy

Comparison Geometry With L1-Norms of Ricci Curvature

2006 ◽  
Vol 49 (1) ◽  
pp. 152-160
Author(s):  
Jong-Gug Yun
Keyword(s):  
Ricci Curvature ◽  
Comparison Theorem ◽  
Sphere Theorem ◽  
Volume Comparison ◽  
Comparison Geometry

AbstractWe investigate the geometry of manifolds with bounded Ricci curvature in L1-sense. In particular, we generalize the classical volume comparison theorem to our situation and obtain a generalized sphere theorem.

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.

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