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Ricci and flag curvatures in Finsler geometry
 In A Sampler of Riemann–Finsler Geometry, Mathematical Sciences Research Institute Publications
, 2004
"... 1.1. Finsler metrics 199 1.2. Flag curvature 207 ..."
Towards the Poincare conjecture and the classification of 3–manifolds,
 Notices Amer. Math. Soc.
, 2003
"... ..."
Geometrization of 3manifolds via the Ricci flow
 Notices of the AMS
, 2004
"... The classification of closed surfaces is a milestone in the development of topology, so much so that it is now taught to most mathematics undergraduates as an introduction to topology. Since the solution ..."
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The classification of closed surfaces is a milestone in the development of topology, so much so that it is now taught to most mathematics undergraduates as an introduction to topology. Since the solution
ANCIENT SOLUTIONS TO KÄHLERRICCI FLOW
, 2005
"... In this paper, we prove that any nonflat ancient solution to KählerRicci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also classify all complete gradient shrinking solitons with nonnegative bisectional curvature. Both results generalize the correspondi ..."
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In this paper, we prove that any nonflat ancient solution to KählerRicci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also classify all complete gradient shrinking solitons with nonnegative bisectional curvature. Both results generalize the corresponding earlier results of Perelman in [P1] and [P2]. The results then are applied to study the geometry and function theory of complete Kähler manifolds with nonnegative bisectional curvature via KählerRicci flow. A compactness result on ancient solutions to KählerRicci flow is also obtained.
HamiltonPerelman’s Proof of the Poincaré Conjecture and The Geometrization Conjecture
, 2006
"... In this paper, we provide an essentially selfcontained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of threemanifolds. In particular, we give a detailed exposition of a complete pro ..."
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In this paper, we provide an essentially selfcontained and detailed account of the fundamental works of Hamilton and the recent breakthrough of Perelman on the Ricci flow and their application to the geometrization of threemanifolds. In particular, we give a detailed exposition of a complete proof of the Poincaré conjecture due to Hamilton and Perelman.
Renormalization group flows and continual Lie algebras”, JHEP 0308
, 2003
"... We study the renormalization group flows of twodimensional metrics in sigma models using the oneloop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the worldsheet lengt ..."
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We study the renormalization group flows of twodimensional metrics in sigma models using the oneloop beta functions, and demonstrate that they provide a continual analogue of the Toda field equations in conformally flat coordinates. In this algebraic setting, the logarithm of the worldsheet length scale, t, is interpreted as Dynkin parameter on the root system of a novel continual Lie algebra, denoted by G(d/dt; 1), with antisymmetric Cartan kernel K(t, t ′ ) = δ ′ (t − t ′); as such, it coincides with the Cartan matrix of the superalgebra sl(NN + 1) in the large N limit. The resulting Toda field equation is a nonlinear generalization of the heat equation, which is integrable in target space and shares the same dissipative properties in time, t. We provide the general solution of the renormalization group flows in terms of free fields, via Bäcklund transformations, and present some simple examples that illustrate the validity of their formal power series expansion in terms of algebraic data. We study in detail the sausage model that arises as geometric deformation of the O(3) sigma model, and give a new interpretation to its ultraviolet limit by gluing together two copies of Witten’s twodimensional black hole in
A CANONICAL COMPATIBLE METRIC FOR GEOMETRIC STRUCTURES ON NILMANIFOLDS
, 2004
"... Abstract. Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost ’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of ..."
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Abstract. Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost ’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci ’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants. Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory. 1.
Type II extinction profile of maximal solutions to the Ricci flow
 in R 2 . arXiv:math.AP/0606288
"... Abstract. We consider the initial value problem ut = ∆log u, u(x,0) = u0(x) ≥ 0 in R 2, corresponding to the Ricci flow, namely conformal evolution of the metric u(dx2 1 + dx2 2) by Ricci curvature. It is well known that the maximal (complete) solution u vanishes identically after time T = 1 4π R2 ..."
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Abstract. We consider the initial value problem ut = ∆log u, u(x,0) = u0(x) ≥ 0 in R 2, corresponding to the Ricci flow, namely conformal evolution of the metric u(dx2 1 + dx2 2) by Ricci curvature. It is well known that the maximal (complete) solution u vanishes identically after time T = 1 4π R2 u0. Assuming that u0 is compactly supported we describe precisely the Type II vanishing behavior of u at time T: we show the existence of an inner region with exponentially fast vanishing profile, which is, up to proper scaling, a soliton cigar solution, and the existence of an outer region of persistence of a logarithmic cusp. This is the only Type II singularity which has been shown to exist, so far, in the Ricci Flow in any dimension. It recovers rigorously formal asymptotics derived by J.R. King [28].