Results 1  10
of
63
On the geometry of metric measure spaces
 II, Acta Math
"... We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Amo ..."
Abstract

Cited by 97 (4 self)
 Add to MetaCart
We introduce and analyze lower (’Ricci’) curvature bounds Curv(M, d,m) ≥ K for metric measure spaces (M, d,m). Our definition is based on convexity properties of the relative entropy Ent(.m) regarded as a function on the L2Wasserstein space of probability measures on the metric space (M, d). Among others, we show that Curv(M, d,m) ≥ K implies estimates for the volume growth of concentric balls. For Riemannian manifolds, Curv(M, d,m) ≥ K if and only if RicM (ξ, ξ) ≥ K · ξ2 for all ξ ∈ TM. The crucial point is that our lower curvature bounds are stable under an appropriate notion of Dconvergence of metric measure spaces. We define a complete and separable metric D on the family of all isomorphism classes of normalized metric measure spaces. The metric D has a natural interpretation, based on the concept of optimal mass transportation. We also prove that the family of normalized metric measure spaces with doubling constant ≤ C is closed under Dconvergence. Moreover, the family of normalized metric measure spaces with doubling constant ≤ C and radius ≤ R is compact under Dconvergence. 1
On the structure of spaces with Ricci curvature bounded below. I
 J. DIFFERENTIAL GEOM
, 1997
"... ..."
(Show Context)
Collapsing riemannian manifolds while keeping their curvature bounded
 I, J. Differential Geometry
, 1986
"... This is the second of two papers concerned with the situation in which the injectivity radius at certain points of a riemannian manifold is "small" compared to the curvature. In Part I [3], we introduced the concept of an Fstructure of positive ..."
Abstract

Cited by 93 (5 self)
 Add to MetaCart
(Show Context)
This is the second of two papers concerned with the situation in which the injectivity radius at certain points of a riemannian manifold is "small" compared to the curvature. In Part I [3], we introduced the concept of an Fstructure of positive
Curvature And Symmetry Of Milnor Spheres
 Ann. of Math
"... this paper to also analyze bundles with base CP CP # CP , and S ..."
Abstract

Cited by 87 (18 self)
 Add to MetaCart
(Show Context)
this paper to also analyze bundles with base CP CP # CP , and S
Almost flat manifolds
 J. Differential Geometry
, 1978
"... 1.1. We denote by V a connected ^dimensional complete Riemannian manifold, by d = d(V) the diameter of V, and by c + = c + (V) and c ~ = c~(V), respectively, the upper and lower bounds of the sectional curvature of V. We set c = c(V) = max (  c + 1,  c ~ ). ..."
Abstract

Cited by 56 (1 self)
 Add to MetaCart
1.1. We denote by V a connected ^dimensional complete Riemannian manifold, by d = d(V) the diameter of V, and by c + = c + (V) and c ~ = c~(V), respectively, the upper and lower bounds of the sectional curvature of V. We set c = c(V) = max (  c + 1,  c ~ ).
Scalar curvature and geometrization conjectures for 3manifolds
 in Comparison Geometry (Berkeley 1993–94), MSRI Publications
, 1997
"... Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization ..."
Abstract

Cited by 31 (8 self)
 Add to MetaCart
Abstract. We first summarize very briefly the topology of 3manifolds and the approach of Thurston towards their geometrization. After discussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof.
Diffeomorphism finiteness, positive pinching, and second homotopy
 Geom. Funct. Anal
, 1999
"... ..."
Convergence theorems in Riemannian geometry
 COMPARISON GEOMETRY
, 1997
"... This is a survey on the convergence theory developed rst by Cheeger and Gromov. In their theory one is concerned with the compactness of the class of riemannian manifolds with bounded curvature and lower bound on the injectivity radius. We explain and give proofs of almost all the major results, inc ..."
Abstract

Cited by 25 (2 self)
 Add to MetaCart
This is a survey on the convergence theory developed rst by Cheeger and Gromov. In their theory one is concerned with the compactness of the class of riemannian manifolds with bounded curvature and lower bound on the injectivity radius. We explain and give proofs of almost all the major results, including Anderson's generalizations to the case where all one has is bounded Ricci curvature. The exposition is streamlined by the introduction of a norm for riemannian manifolds, which makes the theory more like that of Holder and Sobolev spaces.
On Stationary Vacuum Solutions To The Einstein Equations
, 1999
"... this paper is that in fact there are no such nontrivial stationary spacetimes; this of course places the physical reasoning above on stronger footing ..."
Abstract

Cited by 16 (8 self)
 Add to MetaCart
(Show Context)
this paper is that in fact there are no such nontrivial stationary spacetimes; this of course places the physical reasoning above on stronger footing