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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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... 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.
Combinatorial Ricci flows on surfaces
 JOURNAL OF DIFFERENTIAL GEOMETRY
, 2003
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Global existence and convergence of Yamabe flow
 Centre for Mathematical Sciences, Zhejiang University, Hangzhou 310027, China. Weimin Sheng: Department of Mathematics, Zhejiang University, Hangzhou
, 1994
"... Let Mn be a closed connected manifold of dimension n> 3 and [g0] a given conformal class of metrics on M. We consider the (normalized) total scalar curvature functional S on [gQ], S{g)= ..."
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Cited by 50 (0 self)
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Let Mn be a closed connected manifold of dimension n> 3 and [g0] a given conformal class of metrics on M. We consider the (normalized) total scalar curvature functional S on [gQ], S{g)=
THE KÄHLERRICCI FLOW AND THE ¯∂ OPERATOR ON VECTOR FIELDS
"... The limiting behavior of the normalized KählerRicci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is sh ..."
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Cited by 41 (12 self)
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The limiting behavior of the normalized KählerRicci flow for manifolds with positive first Chern class is examined under certain stability conditions. First, it is shown that if the Mabuchi Kenergy is bounded from below, then the scalar curvature converges uniformly to a constant. Second, it is shown that if the Mabuchi Kenergy is bounded from below and if the lowest positive eigenvalue of the ¯ ∂ † ¯ ∂ operator on smooth vector fields is bounded away from 0 along the flow, then the metrics converge exponentially fast in C∞ to a KählerEinstein metric.
The KählerRicci flow through singularities
"... Abstract We prove the existence and uniqueness of the weak KählerRicci flow on projective varieties with log terminal singularities. It is also shown that the weak KählerRicci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an ..."
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Cited by 40 (9 self)
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Abstract We prove the existence and uniqueness of the weak KählerRicci flow on projective varieties with log terminal singularities. It is also shown that the weak KählerRicci flow can be uniquely continued through divisorial contractions and flips if they exist. We then propose an
Discrete Surface Ricci Flow
 SUBMITTED TO IEEE TVCG
"... This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conform ..."
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Cited by 40 (22 self)
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This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conformal (anglepreserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton’s method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.
Recent developments on the Ricci flow
, 1998
"... This article reports recent developments of the research on Hamilton’s Ricci flow and its applications. ..."
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Cited by 39 (3 self)
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This article reports recent developments of the research on Hamilton’s Ricci flow and its applications.
Greedy routing with guaranteed delivery using ricci flows
 In Proc. of the 8th International Symposium on Information Processing in Sensor Networks (IPSN’09
, 2009
"... Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In parti ..."
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Cited by 39 (17 self)
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Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with nontriangular faces as holes, by either using the nodes ’ location or using a landmarkbased scheme without node location. The conformal map is computed with Ricci flow such that all the nontriangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossipstyle computation. The method applies to both unit disk graph models and quasiunit disk graph models. Simulation results are presented for these scenarios.