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An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach
 Applied and Computational Harmonic Analysis, 2007. doi: 10.1016/j.acha.2006.07.004. URL http://www.mat.univie.ac.at/~michor/curveshamiltonian.pdf
"... Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2metrics weighted by expressions in length a ..."
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Cited by 74 (25 self)
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Abstract. Here shape space is either the manifold of simple closed smooth unparameterized curves in R 2 or is the orbifold of immersions from S 1 to R 2 modulo the group of diffeomorphisms of S 1. We investige several Riemannian metrics on shape space: L 2metrics weighted by expressions in length and curvature. These include a scale invariant metric and a Wasserstein type metric which is sandwiched between two lengthweighted metrics. Sobolev metrics of order n on curves are described. Here the horizontal projection of a tangent field is given by a pseudodifferential operator. Finally the metric induced from the Sobolev metric on the group of diffeomorphisms on R 2 is treated. Although the quotient metrics are all given by pseudodifferential operators, their inverses are given by convolution with smooth kernels. We are able to prove local existence and uniqueness of solution to the geodesic equation for both kinds of Sobolev metrics. We are interested in all conserved quantities, so the paper starts with the Hamiltonian setting and computes conserved momenta and geodesics in general on the space of immersions. For each metric we compute the geodesic equation on shape space. In the end we sketch in some examples the differences between these metrics.
Optimal mass transportation and Mather theory
 JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
, 2005
"... We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz ..."
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Cited by 54 (4 self)
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We study the Monge transportation problem when the cost is the action associated to a Lagrangian function on a compact manifold. We show that the transportation can be interpolated by a Lipschitz lamination. We describe several direct variational problems the minimizers of which are these Lipschitz laminations. We prove the existence of an optimal transport map when the transported measure is absolutely continuous. We explain the relations with Mather’s minimal measures.
RICCI FLOW, ENTROPY AND OPTIMAL TRANSPORTATION
"... Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ..."
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Cited by 36 (1 self)
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Abstract. Let a smooth family of Riemannian metrics g(τ) satisfy the backwards Ricci flow equation on a compact oriented ndimensional manifold M. Suppose two families of normalized nforms ω(τ) ≥ 0 and ˜ω(τ) ≥ 0 satisfy the forwards (in τ) heat equation on M generated by the connection Laplacian ∆g(τ). If these nforms represent two evolving distributions of particles over M, the minimum rootmeansquare distance W2(ω(τ), ˜ω(τ), τ) to transport the particles of ω(τ) onto those of ˜ω(τ) is shown to be nonincreasing as a function of τ, without sign conditions on the curvature of (M, g(τ)). Moreover, this contractivity property is shown to characterize supersolutions to the Ricci flow.
Modeling solutions with jumps for rateindependent systems on metric spaces
 Discrete Contin. Dyn. Syst. Ser. A
"... (Communicated by Giuseppe Buttazzo) Abstract. Rateindependent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of ..."
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Cited by 34 (13 self)
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(Communicated by Giuseppe Buttazzo) Abstract. Rateindependent systems allow for solutions with jumps that need additional modeling. Here we suggest a formulation that arises as limit of viscous regularization of the solutions in the extended state space. Hence, our parametrized metric solutions of a rateindependent system are absolutely continuous mappings from a parameter interval into the extended state space. Jumps appear as generalized gradient flows during which the time is constant. The closely related notion of BV solutions is developed afterwards. Our approach is based on the abstract theory of gradient flows in metric spaces, and comparison with other notions of solutions is given. 1. Introduction. Rateindependent
Free boundaries in optimal transport and MongeAmpère obstacle problems
 ANN MATH
, 2010
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NONSMOOTH CALCULUS
"... Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singu ..."
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Cited by 25 (0 self)
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Abstract. We survey recent advances in analysis and geometry, where first order differential analysis has been extended beyond its classical smooth settings. Such studies have applications to geometric rigidity questions, but are also of intrinsic interest. The transition from smooth spaces to singular spaces where calculus is possible parallels the classical development from smooth functions to functions with weak or generalized derivatives. Moreover, there is a new way of looking at the classical geometric theory of Sobolev functions that is useful in more general contexts. 1.
FROM A LARGEDEVIATIONS PRINCIPLE TO THE WASSERSTEIN GRADIENT FLOW: A NEW MICROMACRO PASSAGE
, 2010
"... We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h> 0, a largedeviations rate functional Jh characterizes the behaviour of the particle system at t = h in terms of the initial ..."
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Cited by 22 (16 self)
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We study the connection between a system of many independent Brownian particles on one hand and the deterministic diffusion equation on the other. For a fixed time step h> 0, a largedeviations rate functional Jh characterizes the behaviour of the particle system at t = h in terms of the initial distribution at t = 0. For the diffusion equation, a single step in the timediscretized entropyWasserstein gradient flow is characterized by the minimization of a functional Kh. We establish a new connection between these systems by proving that Jh and Kh are equal up to second order in h as h → 0. This result gives a microscopic explanation of the origin of the entropyWasserstein gradient flow formulation of the diffusion equation. Simultaneously, the limit passage presented here gives a physically natural description of the underlying particle system by describing it as an entropic gradient flow.
A twoscale approach to logarithmic Sobolev inequalities and the hydrodynamic limit
, 2008
"... We consider the coarsegraining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to ..."
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Cited by 21 (2 self)
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We consider the coarsegraining of a lattice system with continuous spin variable. In the first part, two abstract results are established: sufficient conditions for a logarithmic Sobolev inequality with constants independent of the dimension (Theorem 3) and sufficient conditions for convergence to the hydrodynamic limit (Theorem 8). In the second part, we use the abstract results to treat a specific example, namely the Kawasaki dynamics with Ginzburg–
EXISTENCE AND UNIQUENESS OF SOLUTIONS TO AN AGGREGATION EQUATION WITH DEGENERATE DIFFUSION
"... Abstract. We present an energymethodsbased proof of the existence and uniqueness of solutions of a nonlocal aggregation equation with degenerate diffusion. The equation we study is relevant to models of biological aggregation. ..."
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Cited by 19 (3 self)
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Abstract. We present an energymethodsbased proof of the existence and uniqueness of solutions of a nonlocal aggregation equation with degenerate diffusion. The equation we study is relevant to models of biological aggregation.