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.

scholarly journals A note on complete hypersurfaces of non-positive Ricci curvature

1983 ◽  
Vol 28 (3) ◽  
pp. 339-342 ◽  
Author(s):  
G.H. Smith
Keyword(s):  
Euclidean Space ◽  
Recent Result ◽  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Positive Constant ◽  
Constant Sectional Curvature ◽  
Positive Ricci Curvature ◽  
Simply Connected ◽  
Complete Hypersurfaces

In this note we point out that a recent result of Leung concerning hypersurfaces of a Euclidean space has a simple generalisation to hypersurfaces of complete simply-connected Riemannian manifolds of non-positive constant sectional curvature.

A note on Ricci flow with Ricci curvature bounded below

2017 ◽  
Vol 2017 (726) ◽  
Author(s):  
Xiuxiong Chen ◽  
Fang Yuan
Keyword(s):  
Ricci Curvature ◽  
Ricci Flow ◽  
Ricci Flows ◽  
Bounded Below

AbstractIn this note, we will prove that given a sequence of Ricci flows

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 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 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 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 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.

HARMONIC MORPHISMS WITH ONE-DIMENSIONAL FIBRES

1999 ◽  
Vol 10 (04) ◽  
pp. 457-501 ◽  
Author(s):  
RADU PANTILIE
Keyword(s):  
Sectional Curvature ◽  
Ricci Curvature ◽  
Riemannian Manifolds ◽  
Constant Sectional Curvature ◽  
Harmonic Morphisms ◽  
One Dimensional ◽  
Killing Fields ◽  
Dimension One ◽  
Noncompact Riemannian Manifolds ◽  
Integral Formulae

We study harmonic morphisms by placing them into the context of conformal foliations. Most of the results we obtain hold for fibres of dimension one and codomains of dimension not equal to two. We consider foliations which produce harmonic morphisms on both compact and noncompact Riemannian manifolds. By using integral formulae, we prove an extension to one-dimensional foliations which produce harmonic morphisms of the well-known result of S. Bochner concerning Killing fields on compact Riemannian manifolds with nonpositive Ricci curvature. From the noncompact case, we improve a result of R. L. Bryant[9] regarding harmonic morphisms with one-dimensional fibres defined on Riemannian manifolds of dimension at least four with constant sectional curvature. Our method gives an entirely new and geometrical proof of Bryant's result. The concept of homothetic foliation (or, more generally, homothetic distribution) which we introduce, appears as a useful tool both in proofs and in providing new examples of harmonic morphisms, with fibres of any dimension.

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