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On the geometry of metric measure spaces
 II, ACTA MATH
, 2004
"... 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 ..."
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Cited by 247 (9 self)
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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.
Manifolds with positive curvature operator are space forms
 ANN. OF MATH
"... ... that a compact threemanifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact fourmanifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for ..."
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Cited by 115 (2 self)
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... that a compact threemanifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact fourmanifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact fourmanifolds with 2positive curvature operators [Che]. Recall that a curvature operator is called 2positive, if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken [Hu] described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally, we show the following Theorem 1. On a compact manifold the normalized Ricci flow evolves a Riemannian metric with 2positive curvature operator to a limit metric with constant sectional curvature. The theorem is known in dimensions below five [H3], [H1], [Che]. Our proof works in dimensions above two: we only use Hamilton’s maximum principle and Klingenberg’s injectivity radius estimate for quarter pinched manifolds. Since in dimensions above two a quarter pinched orbifold is covered by a manifold (see Proposition 5.2), our proof carries over to orbifolds. This is no longer true in dimension two. In the manifold case it is known that the normalized Ricci flow converges to a metric of constant curvature for any initial metric [H3], [Cho]. However, there exist twodimensional orbifolds with positive sectional curvature which are not covered by a manifold. On such orbifolds the Ricci flow converges to a nontrivial Ricci soliton [CW]. Let us mention that a 2positive curvature operator has positive isotropic curvature. Micallef and Moore [MM] showed that a simply connected compact manifold with positive isotropic curvature is a homotopy sphere. However, their techniques do not allow to get restrictions for the fundamental groups or the differentiable structure of the underlying manifold.
Membranes at Quantum Criticality
, 2009
"... We propose a quantum theory of membranes designed such that the groundstate wavefunction of the membrane with compact spatial topology Σh reproduces the partition function of the bosonic string on worldsheet Σh. The construction involves worldvolume matter at quantum criticality, described in the ..."
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Cited by 106 (0 self)
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We propose a quantum theory of membranes designed such that the groundstate wavefunction of the membrane with compact spatial topology Σh reproduces the partition function of the bosonic string on worldsheet Σh. The construction involves worldvolume matter at quantum criticality, described in the simplest case by Lifshitz scalars with dynamical critical exponent z = 2. This matter system must be coupled to a novel theory of worldvolume gravity, also exhibiting quantum criticality with z = 2. We first construct such a nonrelativistic “gravity at a Lifshitz point ” with z = 2 in D + 1 spacetime dimensions, and then specialize to the critical case of D = 2 suitable for the membrane worldvolume. We also show that in the secondquantized framework, the string partition function is reproduced if the spacetime ground state takes the form of a BoseEinstein condensate of membranes in
Strong uniqueness of the Ricci flow
 arXiv:0706.3081. HUAIDONG CAO
"... In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow onR 3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t)≡E. 1 ..."
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Cited by 92 (0 self)
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In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow onR 3, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t)≡E. 1
Comparison Geometry for the BakryEmery Ricci tensor
"... For Riemannian manifolds with a measure (M, g, e −f dvolg) we prove mean curvature and volume comparison results when the ∞BakryEmery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extension ..."
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Cited by 77 (7 self)
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For Riemannian manifolds with a measure (M, g, e −f dvolg) we prove mean curvature and volume comparison results when the ∞BakryEmery Ricci tensor is bounded from below and f is bounded or ∂rf is bounded from below, generalizing the classical ones (i.e. when f is constant). This leads to extensions of many theorems for Ricci curvature bounded below to the BakryEmery Ricci tensor. In particular, we give extensions of all of the major comparison theorems when f is bounded. Simple examples show the bound on f is necessary for these results.
Dark Energy from Structure  A Status Report
 GEN. REL. GRAV., DARK ENERGY SPECIAL ISSUE
, 2007
"... The effective evolution of an inhomogeneous universe model in any theory of gravitation may be described in terms of spatially averaged variables. In Einstein’s theory, restricting attention to scalar variables, this evolution can be modeled by solutions of a set of Friedmann equations for an effe ..."
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Cited by 58 (9 self)
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The effective evolution of an inhomogeneous universe model in any theory of gravitation may be described in terms of spatially averaged variables. In Einstein’s theory, restricting attention to scalar variables, this evolution can be modeled by solutions of a set of Friedmann equations for an effective volume scale factor, with matter and backreaction source terms. The latter can be represented by an effective scalar field (‘morphon field’) modeling Dark Energy. The present work provides an overview over the Dark Energy debate in connection with the impact of inhomogeneities, and formulates strategies for a comprehensive quantitative evaluation of backreaction effects both in theoretical and observational cosmology. We recall the basic steps of a description of backreaction effects in relativistic cosmology that lead to refurnishing the standard cosmological equations, but also lay down a number of challenges and unresolved issues in connection with their observational interpretation. The present status of this subject is intermediate: we have a good qualitative understanding of backreaction effects pointing to a global instability of the standard