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449
Heegaard splittings, the virtually Haken Conjecture, and Property (τ)
, 2002
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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.
Entropy and reduced distance for Ricci expanders
 J. Geom. Anal
"... ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij /∂t =−2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expan ..."
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Cited by 40 (6 self)
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ABSTRACT. Perelman has discovered two integral quantities, the shrinker entropy W and the (backward) reduced volume, that are monotone under the Ricci flow ∂gij /∂t =−2Rij and constant on shrinking solitons. Tweaking some signs, we find similar formulae corresponding to the expanding case. The expanding entropy W+ is monotone on any compact Ricci flow and constant precisely on expanders; as in Perelman, it follows from a differential inequality for a Harnacklike quantity for the conjugate heat equation, and leads to functionals µ+ and ν+. The forward reduced volume θ+ is monotone in general and constant exactly on expanders. A natural conjecture asserts that g(t)/t converges as t →∞to a negative Einstein manifold in some weak sense (in particular ignoring collapsing parts). If the limit is known apriori to be smooth and compact, this statement follows easily from any monotone quantity that is constant on expanders; these include vol(g)/t n/2 (Hamilton) and ¯λ (Perelman), as well as our new quantities. In general, we show that, if vol(g) grows like t n/2 (maximal volume growth) then W+, θ+ and ¯λ remain bounded (in their appropriate ways) for all time. We attempt a sharp formulation of the conjecture. Small, large and distant parts of a Ricci flow are known to be modeled by various kinds of Ricci solitons: Steady, shrinking, and expanding. Perelman has discovered monotone quantities
Finite covers of random 3manifolds
, 2005
"... A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed ..."
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Cited by 39 (1 self)
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A 3manifold is Haken if it contains a topologically essential surface. The Virtual Haken Conjecture posits that every irreducible 3manifold with infinite fundamental group has a finite cover which is Haken. In this paper, we study random 3manifolds and their finite covers in an attempt to shed light on this difficult question. In particular, we consider random Heegaard splittings by gluing two handlebodies by the result of a random walk in the mapping class group of a surface. For this model of random 3manifold, we are able to compute the probabilities that the resulting manifolds have finite covers of particular kinds. Our results contrast with the analogous probabilities for groups coming from random balanced presentations, giving quantitative theorems to the effect that 3manifold groups have many more finite quotients than random groups. The next natural question is whether these covers have positive betti number. For abelian covers of a fixed type over 3manifolds of Heegaard genus 2, we show that the probability of positive betti number is 0. In fact, many of these questions boil down to questions about the mapping class group. We are lead to consider the action of mapping class group of a surface Σ on
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.
Volume collapsed threemanifolds with a lower curvature
"... Abstract. In this paper we determine the topology of threedimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small. 1. ..."
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Abstract. In this paper we determine the topology of threedimensional closed orientable Riemannian manifolds with a uniform lower bound of sectional curvature whose volume is sufficiently small. 1.
W.Wylie, On the classification of gradient Ricci solitons
"... Abstract. We show that the only shrinking gradient solitons with vanishing Weyl tensor are quotients of the standard ones S n, S n−1 × R, and R n. This gives a new proof of the HamiltonIveyPerel’man classification of 3dimensional shrinking gradient solitons. We also show that gradient solitons wi ..."
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Cited by 35 (2 self)
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Abstract. We show that the only shrinking gradient solitons with vanishing Weyl tensor are quotients of the standard ones S n, S n−1 × R, and R n. This gives a new proof of the HamiltonIveyPerel’man classification of 3dimensional shrinking gradient solitons. We also show that gradient solitons with constant scalar curvature and suitably decaying Weyl tensor when noncompact are quotients of H n, H n−1 × R, R n, S n−1 × R, or S n. 1.