scholarly journals The structure of spaces with Bakry–Émery Ricci curvature bounded below

2019 ◽  
Vol 2019 (757) ◽  
pp. 1-50 ◽  
Author(s):  
Feng Wang ◽  
Xiaohua Zhu
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Ricci Solitons ◽  
Existence Problem ◽  
Fano Manifold ◽  
Hausdorff Topology ◽  
Limit Space ◽  
Gromov Hausdorff Topology ◽  
Bounded Below ◽  
Monge Ampere

AbstractWe explore the structure of limit spaces of sequences of Riemannian manifolds with Bakry–Émery Ricci curvature bounded below in the Gromov–Hausdorff topology. By extending the techniques established by Cheeger and Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at a point of the limit space is a metric cone. We also analyze the singular structure of the limit space as in a paper of Cheeger, Colding and Tian. Our results will be applied to study the limit spaces for a sequence of Kähler metrics arising from solutions of certain complex Monge–Ampère equations for the existence problem of Kähler–Ricci solitons on a Fano manifold via the continuity method.

scholarly journals On dimensions of tangent cones in limit spaces with lower Ricci curvature bounds

2018 ◽  
Vol 2018 (742) ◽  
pp. 263-280
Author(s):  
Vitali Kapovitch ◽  
Nan Li
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Tangent Cones ◽  
Hölder Continuous ◽  
Hausdorff Topology ◽  
Curvature Bounds ◽  
Scale Measure ◽  
Gromov Hausdorff Topology ◽  
Bounded Below

Abstract We show that if X is a limit of n-dimensional Riemannian manifolds with Ricci curvature bounded below and γ is a limit geodesic in X, then along the interior of γ same scale measure metric tangent cones {T_{\gamma(t)}X} are Hölder continuous with respect to measured Gromov–Hausdorff topology and have the same dimension in the sense of Colding–Naber.

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 On the limit behavior of metrics in the continuity method for the Kähler–Einstein problem on a toric Fano manifold

2012 ◽  
Vol 148 (6) ◽  
pp. 1985-2003 ◽  
Author(s):  
Chi Li
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Limit Behavior ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Continuity Method ◽  
Type Singularity ◽  
Moment Polytope ◽  
The Moment ◽  
Monge Ampere

AbstractThis work is a continuation of the author’s previous paper [Greatest lower bounds on the Ricci curvature of toric Fano manifolds, Adv. Math. 226 (2011), 4921–4932]. On any toric Fano manifold, we discuss the behavior of the limit metric of a sequence of metrics which are solutions to a continuity family of complex Monge–Ampère equations in the Kähler–Einstein problem. We show that the limit metric satisfies a singular complex Monge–Ampère equation. This gives a conic-type singularity for the limit metric. Information on conic-type singularities can be read off from the geometry of the moment polytope.

scholarly journals Sharp gradient estimates for eigenfunctions on Riemannian manifolds

1998 ◽  
Vol 57 (2) ◽  
pp. 253-259 ◽  
Author(s):  
Albert Borbély
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Gradient Estimates ◽  
Bounded Below ◽  
Complete Riemannian Manifolds

Sharp gradient estimates are derived for positive eigenfunctions on complete Riemannian manifolds with Ricci curvature bounded below.

scholarly journals Greatest lower bounds on the Ricci curvature of Fano manifolds

2010 ◽  
Vol 147 (1) ◽  
pp. 319-331 ◽  
Author(s):  
Gábor Székelyhidi
Keyword(s):  
Ricci Curvature ◽  
Lower Bounds ◽  
Einstein Metrics ◽  
Upper Bounds ◽  
Fano Manifold ◽  
Fano Manifolds ◽  
Existence Time ◽  
Kähler Einstein Metrics ◽  
Bounded Below ◽  
Kahler Metric

AbstractOn a Fano manifoldMwe study the supremum of the possibletsuch that there is a Kähler metricω∈c1(M) with Ricci curvature bounded below byt. This is shown to be the same as the maximum existence time of Aubin’s continuity path for finding Kähler–Einstein metrics. We show that onP2blown up in one point this supremum is 6/7, and we give upper bounds for other manifolds.

scholarly journals A Schwarz lemma for complete Riemannian manifolds

1997 ◽  
Vol 55 (3) ◽  
pp. 513-515
Author(s):  
Leung-Fu Cheung ◽  
Pui-Fai Leung
Keyword(s):  
Ricci Curvature ◽  
Distance Function ◽  
Riemannian Manifolds ◽  
Conformal Mappings ◽  
Schwarz Lemma ◽  
Partial Result ◽  
Bounded Below ◽  
Complete Riemannian Manifolds

We prove a Schwarz Lemma for conformal mappings between two complete Riemannian manifolds when the domain manifold has Ricci curvature bounded below in terms of its distance function. This gives a partial result to a conjecture of Chua.

scholarly journals Collapsing Geometry with Ricci Curvature Bounded Below and Ricci Flow Smoothing

2020 ◽  
Author(s):  
Shaosai Huang ◽  
◽  
Xiaochun Rong ◽  
Bing Wang ◽  
◽  
...  
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Ricci Flow ◽  
Smoothing Techniques ◽  
Ricci Flat ◽  
Recent Developments ◽  
Open Question ◽  
Bounded Below

We survey some recent developments in the study of collapsing Riemannian manifolds with Ricci curvature bounded below, especially the locally bounded Ricci covering geometry and the Ricci flow smoothing techniques. We then prove that if a Calabi-Yau manifold is sufficiently volume collapsed with bounded diameter and sectional curvature, then it admits a Ricci-flat Kähler metrictogether with a compatible pure nilpotent Killing structure: this is related to an open question of Cheeger, Fukaya and Gromov.

ON THE COMPACTNESS OF MANIFOLDS

2003 ◽  
Vol 06 (supp01) ◽  
pp. 29-38 ◽  
Author(s):  
XUE-MEI LI ◽  
FENG-YU WANG
Keyword(s):  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Constant ◽  
Riemannian Metric ◽  
Sobolev Inequalities ◽  
Functional Inequalities ◽  
The Family ◽  
Bounded Below

It is believed that the family of Riemannian manifolds with negative curvatures is much richer than that with positive curvatures. In fact there are many results on the obstruction of furnishing a manifold with a Riemannian metric whose curvature is positive. In particular any manifold admitting a Riemannian metric whose Ricci curvature is bounded below by a positive constant must be compact. Here we investigate such obstructions in terms of certain functional inequalities which can be considered as generalized Poincaré or log-Sobolev inequalities. A result of Saloff-Coste is extended.

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