scholarly journals On Measure Contraction Property without Ricci Curvature Lower Bound

2015 ◽  
Vol 44 (1) ◽  
pp. 27-41 ◽  
Author(s):  
Paul W. Y. Lee
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Contraction Property ◽  
Measure Contraction Property

scholarly journals Inscribed Radius Bounds for Lower Ricci Bounded Metric Measure Spaces with Mean Convex Boundary

2020 ◽  
Author(s):  
Annegret Burtscher ◽  
◽  
Christian Ketterer ◽  
Robert J. McCann ◽  
Eric Woolgar ◽  
...  
Keyword(s):  
Lower Bound ◽  
Mean Curvature ◽  
Measure Space ◽  
Riemannian Curvature ◽  
Metric Measure Spaces ◽  
Contraction Property ◽  
Measure Spaces ◽  
Generalized Mean Curvature ◽  
Measure Contraction Property ◽  
Mean Convex

Consider an essentially nonbranching metric measure space with the measure contraction property of Ohta and Sturm, or with a Ricci curvature lower bound in the sense of Lott, Sturm and Villani. We prove a sharp upper bound on the inscribed radius of any subset whose boundary has a suitably signed lower bound on its generalized mean curvature. This provides a nonsmooth analog to a result of Kasue (1983) and Li (2014). We prove a stability statement concerning such bounds and - in the Riemannian curvature-dimension (RCD) setting - characterize the cases of equality.

scholarly journals Poincaré inequality for abstract spaces

2005 ◽  
Vol 71 (2) ◽  
pp. 193-204 ◽  
Author(s):  
Alireza Ranjbar-Motlagh
Keyword(s):  
Ricci Curvature ◽  
Poincaré Inequality ◽  
Strong Version ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Poincare Inequality ◽  
Contraction Property ◽  
Measure Spaces ◽  
Measure Contraction Property ◽  
Abstract Spaces

The Poincaré inequality is generalised to metric-measure spaces which support a strong version of the doubling condition. This generalises the Poincaré inequality for manifolds whose Ricci curvature is bounded from below and metric-measure spaces which satisfy the measure contraction property.

Exit radii of submanifolds from cylindrical domains in warped product manifolds

2015 ◽  
Vol 145 (3) ◽  
pp. 559-569
Author(s):  
Hironori Kumura
Keyword(s):  
Lower Bound ◽  
Bounded Domain ◽  
Ricci Curvature ◽  
Mean Curvature ◽  
Warped Product ◽  
Product Manifold ◽  
Warped Product Manifold ◽  
Cylindrical Domains ◽  
The Mean

Let UB(p0; ρ1) × f MV be a cylindrically bounded domain in a warped product manifold := MB × fMV and let M be an isometrically immersed submanifold in . The purpose of this paper is to provide explicit radii of the geodesic balls of M which first exit from UB(p0; ρ1) × fMV for the case in which the mean curvature of M is sufficiently small and the lower bound of the Ricci curvature of M does not diverge to –∞ too rapidly at infinity.

scholarly journals Distortion Coefficients of the $$\alpha $$-Grushin Plane

2022 ◽  
Vol 32 (3) ◽  
Author(s):  
Samuël Borza
Keyword(s):  
Trigonometric Functions ◽  
Contraction Property ◽  
Grushin Plane ◽  
Measure Contraction Property ◽  
Property Condition

AbstractWe compute the distortion coefficients of the $$\alpha $$ α -Grushin plane. They are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a measure contraction property condition for the generalised Grushin planes is suggested.

scholarly journals On low-dimensional Ricci limit spaces

2013 ◽  
Vol 209 ◽  
pp. 1-22 ◽  
Author(s):  
Shouhei Honda
Keyword(s):  
Lower Bound ◽  
Hausdorff Dimension ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Limit Space ◽  
Low Dimensional ◽  
Complete Riemannian Manifolds

AbstractWe call a Gromov–Hausdorff limit of complete Riemannian manifolds with a lower bound of Ricci curvature a Ricci limit space. Furthermore, we prove that any Ricci limit space has integral Hausdorff dimension, provided that its Hausdorff dimension is not greater than 2. We also classify 1-dimensional Ricci limit spaces.

scholarly journals A note on bounded variation and heat semigroup on Riemannian manifolds

2007 ◽  
Vol 76 (1) ◽  
pp. 155-160 ◽  
Author(s):  
A. Carbonaro ◽  
G. Mauceri
Keyword(s):  
Lower Bound ◽  
Euclidean Space ◽  
Heat Kernel ◽  
Ricci Curvature ◽  
Bounded Variation ◽  
Riemannian Manifolds ◽  
Heat Semigroup ◽  
Functions Of Bounded Variation ◽  
Fixed Radius ◽  
Lower Bound Estimate

In a recent paper Miranda Jr., Pallara, Paronetto and Preunkert have shown that the classical De Giorgi's heat kernel characterisation of functions of bounded variation on Euclidean space extends to Riemannian manifolds with Ricci curvature bounded from below and which satisfy a uniform lower bound estimate on the volume of geodesic balls of fixed radius. We give a shorter proof of the same result assuming only the lower bound on the Ricci curvature.

scholarly journals RICCI LOWER BOUND FOR KÄHLER–RICCI FLOW

2014 ◽  
Vol 16 (02) ◽  
pp. 1350053 ◽  
Author(s):  
ZHOU ZHANG
Keyword(s):  
Lower Bound ◽  
Ricci Curvature ◽  
Ricci Flow ◽  
Finite Time ◽  
Ricci Flows ◽  
Time Singularities

We provide general discussion on the lower bound of Ricci curvature along Kähler–Ricci flows over closed manifolds. The main result is the non-existence of Ricci lower bound for flows with finite time singularities and non-collapsed global volume. As an application, we give examples showing that positivity of Ricci curvature would not be preserved by Ricci flow in general.

scholarly journals Sharp measure contraction property for generalized H-type Carnot groups

2018 ◽  
Vol 20 (06) ◽  
pp. 1750081 ◽  
Author(s):  
Davide Barilari ◽  
Luca Rizzi
Keyword(s):  
Absolute Continuity ◽  
Carnot Group ◽  
Optimal Synthesis ◽  
Carnot Groups ◽  
Contraction Property ◽  
Dimension Formula ◽  
Quadratic Cost ◽  
Measure Contraction Property ◽  
The Absolute

We prove that H-type Carnot groups of rank [Formula: see text] and dimension [Formula: see text] satisfy the [Formula: see text] if and only if [Formula: see text] and [Formula: see text]. The latter integer coincides with the geodesic dimension of the Carnot group. The same result holds true for the larger class of generalized H-type Carnot groups introduced in this paper, and for which we compute explicitly the optimal synthesis. This constitutes the largest class of Carnot groups for which the curvature exponent coincides with the geodesic dimension. We stress that generalized H-type Carnot groups have step 2, include all corank 1 groups and, in general, admit abnormal minimizing curves. As a corollary, we prove the absolute continuity of the Wasserstein geodesics for the quadratic cost on all generalized H-type Carnot groups.

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