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13
RICCI FLOW AND THE METRIC COMPLETION OF THE SPACE OF KÄHLER METRICS
, 2012
"... We consider the space of Kähler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of K ..."
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Cited by 12 (4 self)
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We consider the space of Kähler metrics as a Riemannian submanifold of the space of Riemannian metrics, and study the associated submanifold geometry. In particular, we show that the intrinsic and extrinsic distance functions are equivalent. We also determine the metric completion of the space of Kähler metrics, making contact with recent generalizations of the Calabi–Yau Theorem due to Dinew and Guedj–Zeriahi. As an application, we obtain a new analytic stability criterion for the existence of a Kähler–Einstein metric on a Fano manifold in terms of the Ricci flow and the distance function. We also prove that the Kähler–Ricci flow converges as soon as it converges in the metric sense.
The Completion of the Manifold of Riemannian Metrics
"... Abstract. We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional, orientable manifold with respect to a natural metric called the L 2 metric. The primary motivation for studying this problem comes from Teichmüller theory ..."
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Cited by 9 (5 self)
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Abstract. We give a description of the completion of the manifold of all smooth Riemannian metrics on a fixed smooth, closed, finitedimensional, orientable manifold with respect to a natural metric called the L 2 metric. The primary motivation for studying this problem comes from Teichmüller theory, where similar considerations lead to a completion of the wellknown WeilPetersson metric. We give an application of the main theorem to the completions of Teichmüller space with respect to a class of metrics that generalize the WeilPetersson metric.
SOBOLEV METRICS ON THE MANIFOLD OF ALL RIEMANNIAN METRICS
"... Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellpo ..."
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Cited by 4 (2 self)
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Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as described first by [11]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellposed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of these metrics. 1.
Some recent work in Fréchet geometry
 INTERNATIONAL CONFERENCE ON DIFFERENTIAL GEOMETRY AND
, 2011
"... Fréchet spaces of sections arise naturally as configurations of a physical field. Here some recent work in Fréchet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manif ..."
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Fréchet spaces of sections arise naturally as configurations of a physical field. Here some recent work in Fréchet geometry is briefly reviewed. In particular an earlier result on the structure of second tangent bundles in the finite dimensional case was extended to infinite dimensional Banach manifolds and Fréchet manifolds that could be represented as projective limits of Banach manifolds. This led to further results concerning the characterization of second tangent bundles and differential equations in the more general Fréchet structure needed for applications. A summary is given of recent results on hypercyclicity of
Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics. arXiv
, 1104
"... ar ..."
Elastic shape matching of parameterized surfaces using square root normal fields
 In Proceedings of the 12th European conference on Computer Vision  Volume Part V, ECCV’12
, 2012
"... Abstract. In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. Th ..."
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Abstract. In this paper we define a new methodology for shape analysis of parameterized surfaces, where the main issues are: (1) choice of metric for shape comparisons and (2) invariance to reparameterization. We begin by defining a general elastic metric on the space of parameterized surfaces. The main advantages of this metric are twofold. First, it provides a natural interpretation of elastic shape deformations that are being quantified. Second, this metric is invariant under the action of the reparameterization group. We also introduce a novel representation of surfaces termed square root normal fields or SRNFs. This representation is convenient for shape analysis because, under this representation, a reduced version of the general elastic metric becomes the simple L2 metric. Thus, this transformation greatly simplifies the implementation of our framework. We validate our approach using multiple shape analysis examples for quadrilateral and spherical surfaces. We also compare the current results with those of Kurtek et al. [1]. We show that the proposed method results in more natural shape matchings, and furthermore, has some theoretical advantages over previous methods. 1
Sobolev metrics on the Riemannian manifold of all Riemannian metrics
, 2010
"... Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellpos ..."
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Abstract. On the manifold M(M) of all Riemannian metrics on a compact manifold M one can consider the natural L 2metric as decribed first by [10]. In this paper we consider variants of this metric which in general are of higher order. We derive the geodesic equations, we show that they are wellposed under some conditions and induce a locally diffeomorphic geodesic exponential mapping. We give a condition when Ricci flow is a gradient flow for one of this metrics. 1.
SCALAR CURVATURE AND QCURVATURE OF RANDOM METRICS.
"... Abstract. We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. We next consider analogous questions for the scalar curvature in dimension n> 2, and for the Qcurvature of random Riemannia ..."
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Abstract. We study Gauss curvature for random Riemannian metrics on a compact surface, lying in a fixed conformal class; our questions are motivated by comparison geometry. We next consider analogous questions for the scalar curvature in dimension n> 2, and for the Qcurvature of random Riemannian metrics. 1.
A review of some recent work on hypercyclicity
"... Even linear operators on infinitedimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of applications. In particular, hypercyclicity is an essentially infinited ..."
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Even linear operators on infinitedimensional spaces can display interesting dynamical properties and yield important links among functional analysis, differential and global geometry and dynamical systems, with a wide range of applications. In particular, hypercyclicity is an essentially infinitedimensional property, when iterations of the operator generate a dense subspace. A Fréchet space admits a hypercyclic operator if and only if it is separable and infinitedimensional. However, by considering the semigroups generated by multiples of operators, it is possible to obtain hypercyclic behaviour on finite dimensional spaces. This article gives a brief review of some recent work on hypercyclicity of operators on Banach, Hilbert and Fréchet spaces.