On a Formula for the L2 Wasserstein Metric between Measures on Euclidean and Hilbert Spaces

For a separable metric space (X, d) L^p Wasserstein metrics between probability measures μ and v on X are defined by where the infimum is taken over all probability measures η on X × X with marginal distributions μ and v, respectively. After mentioning some basic properties of these metrics as well as explicit formulae for X = R a formula for the L² Wasserstein metric with X = Rⁿ will be cited from [5], [9], and [21] and proved for any two probability measures of a family of elliptically contoured distributions. Finally this result will be generalized for Gaussian measures to the case of a separable Hilbert space.     

Bach-flat asymptotically locally Euclidean metrics

We obtain a volume growth and curvature decay result for various classes of complete, noncompact Riemannian metrics in dimension 4; in particular our method applies to anti-self-dual or Kähler metrics with zero scalar curvature, and metrics with harmonic curvature. Similar results were obtained for Einstein metrics in [And89], [BKN89], [Tia90], but our analysis differs from the Einstein case in that (1) we consider more generally a fourth order system in the metric, and (2) we do not assume any pointwise Ricci curvature bound.     

Central limit theorems for solutions of the Kac equation: speed of approach to equilibrium in weak metrics

This paper is part of our efforts to show how direct application of probabilistic methods, pertaining to central limit general theory, can enlighten us about the convergence to equilibrium of the solutions of the Kac equation. Here, we consider convergence with respect to the following metrics: Kolmogorov’s uniform metric; 1 and 2 Gini’s dissimilarity indices (widely known as 1 and 2 Wasserstein metrics); χ-weighted metrics. Our main results provide new bounds, or improvements on already well-known ones, for the corresponding distances between the solution of the Kac equation and the limiting Gaussian (Maxwellian) distribution. The study is conducted both under the necessary assumption that initial data have finite energy, without assuming existence of moments of order greater than 2, and under the condition that the (2 + δ)-moment of the initial distribution is finite for some δ > 0.     

Curvature and uniformization

We approach the problem of uniformization of general Riemann surfaces through consideration of the curvature equation, and in particular the problem of constructing Poincaré metrics (i.e., complete metrics of constant negative curvature) by solving the equation Δu-e ^2u=K_o(z) on general open surfaces. A few other topics are discussed, including boundary behavior of the conformal factore ^2u giving the Poincaré metric when the Riemann surface has smoothly bounded compact closure, and also a curvature equation proof of Koebe's disk theorem. Research supported in part by NSF Grant DMS-9971975 and also at MSRI by NSF grant DMS-9701755. Research supported in part by NSF Grant DMS-9877077     

A renormalized index theorem for some complete asymptotically regular metrics: The Gauss-Bonnet theorem

We study the Gauss-Bonnet theorem as a renormalized index theorem for edge metrics. These metrics include the Poincaré-Einstein metrics of the AdS/CFT correspondence and the asymptotically cylindrical metrics of the Atiyah-Patodi-Singer index theorem. We use renormalization to make sense of the curvature integral and the dimensions of the L²-cohomology spaces as well as to carry out the heat equation proof of the index theorem. For conformally compact metrics even mod xm, we show that the finite time supertrace of the heat kernel on conformally compact manifolds renormalizes independently of the choice of special boundary defining function.     

The Abreu equation with degenerated boundary conditions

The Abreu equation is a fully nonlinear 4th order partial differential equation that arises from the study of the extremal metrics on toric manifolds. We study the Dirichlet problem of the Abreu equation with degenerated boundary conditions. The solutions provide the Kähler metrics of constant scalar curvature on the complex torus.     

A physical space approach to wave equation bilinear estimates

Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel’s theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics or connections with non-zero curvature). In this note, we describe an alternative physical space approach which relies on vector fields, energy estimates as well as tube localization, splitting into coarse and fine scales, and induction on scales (in the spirit of Wolff [29], [30]).     

Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities for a nonlinear parabolic equation

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li–Yau differential Harnack inequality for a nonlinear parabolic equation to the constrained trace Chow–Hamilton Harnack inequality for this nonlinear equation with respect to evolving metrics related to the Ricci flow on a 2-dimensional closed manifold. This result can be regarded as a nonlinear version of the previous work of Y. Zheng and the author [J.-Y. Wu, Y. Zheng, Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities on a surface, Arch. Math., 94 (2010) 591–600].     

Dispersion and Strichartz estimates for the Liouville equation

We consider the Liouville equation associated with a metric g of class C² and we prove dispersion and Strichartz estimates for the solution of this equation in terms of geodesics associated with g. We introduce the notion of focusing and dispersive metric to characterize metrics such that the same dispersion estimate as in the Euclidean case holds. To deal with the case of non-trapped long range perturbation of the Euclidean metric, we prove a global velocity moments effect on the solution. In particular, we obtain global in time Strichartz estimates for metrics such that the dispersion estimate is not satisfied.     

Plane Embeddings of Planar Graph Metrics

Embedding metrics into constant-dimensional geometric spaces, such as the Euclidean plane, is relatively poorly understood. Motivated by applications in visualization, ad-hoc networks, and molecular reconstruction, we consider the natural problem of embedding shortest-path metrics of unweighted planar graphs (planar graph metrics) into the Euclidean plane. It is known that, in the special case of shortest-path metrics of trees, embedding into the plane requires distortion in the worst case [M1], [BMMV], and surprisingly, this worst-case upper bound provides the best known approximation algorithm for minimizing distortion. We answer an open question posed in this work and highlighted by Matousek [M3] by proving that some planar graph metrics require distortion in any embedding into the plane, proving the first separation between these two types of graph metrics. We also prove that some planar graph metrics require distortion in any crossing-free straight-line embedding into the plane, suggesting a separation between low-distortion plane embedding and the well-studied notion of crossing-free straight-line planar drawings. Finally, on the upper-bound side, we prove that all outerplanar graph metrics can be embedded into the plane with distortion, generalizing the previous results on trees (both the worst-case bound and the approximation algorithm) and building techniques for handling cycles in plane embeddings of graph metrics.     

Conformal deformations of the Ebin metric and a generalized Calabi metric on the space of Riemannian metrics

We consider geometries on the space of Riemannian metrics conformally equivalent to the widely studied Ebin L²$L^2$ metric. Among these we characterize a distinguished metric that can be regarded as a generalization of Calabi?s metric on the space of Kähler metrics to the space of Riemannian metrics, and we study its geometry in detail. Unlike the Ebin metric, its geodesic equation involves non-local terms, and we solve it explicitly by using a constant of the motion. We then determine its completion, which gives the first example of a metric on the space of Riemannian metrics whose completion is strictly smaller than that of the Ebin metric.     

The Ricci flow as a geodesic on the manifold of Riemannian metrics

The Ricci flow is an evolution equation in the space of Riemannian metrics.A solution for this equation is a curve on the manifold of Riemannian metrics. In this paper we introduce a metric on the manifold of Riemannian metrics such that the Ricci flow becomes a geodesic.We show that the Ricci solitons introduce a special slice on the manifold of Riemannian metrics.     

A global classification result for Randers metrics of scalar curvature on closed manifolds

One important problem in Finsler geometry is that of classifying Finsler metrics of scalar curvature. By investigating the second-order differential equation for a class of Randers metrics with isotropic S$S$-curvature, we give a global classification of these metrics of scalar curvature, generalizing a theorem previously only known in the case of locally projectively flat Randers metrics.     

On Einstein Matsumoto metrics

Einstein metrics are solutions to Einstein field equation in General Relativity containing the Ricci-flat metrics. Einstein Finsler metrics which represent a non-Riemannian stage for the extensions of metric gravity, provide an interesting source of geometric issues and the (α,β)-metric is an important class of Finsler metrics appearing iteratively in physical studies. It is proved that every n-dimensional (n≥3) Einstein Matsumoto metric is a Ricci-flat metric with vanishing S-curvature. The main result can be regarded as a second Schur type Lemma for Matsumoto metrics.     

On spherically symmetric Finsler metrics with vanishing Douglas curvature

We obtain the differential equation that characterizes the spherically symmetric Finsler metrics with vanishing Douglas curvature. By solving this equation, we obtain all the spherically symmetric Douglas metrics. Many explicit examples are included.     

Existence and Uniqueness of Constant Mean Curvature Foliation of Asymptotically Hyperbolic 3-Manifolds

We prove existence and uniqueness of foliations by stable spheres with constant mean curvature for 3-manifolds which are asymptotic to anti-de Sitter–Schwarzschild metrics with positive mass. These metrics arise naturally as spacelike timeslices for solutions of the Einstein equation with a negative cosmological constant.     

Minimizers of a weighted maximum of the Gauss curvature

On a Riemann surface [`(S)]{\overline{\Sigma}} with smooth boundary, we consider Riemannian metrics conformal to a given background metric. Let κ be a smooth, positive function on [`(S)]{\overline{\Sigma}}. If K denotes the Gauss curvature, then the L ^∞-norm of K/κ gives rise to a functional on the space of all admissible metrics. We study minimizers subject to an area constraint. Under suitable conditions, we construct a minimizer with the property that |K|/κ is constant. The sign of K can change, but this happens only on the nodal set of the solution of a linear partial differential equation.     

Parametric versions of Hilbert’s fourth problem

LetH be the class of sufficiently smooth metrics defined on the Euclidean plane for which the geodesics are the usual Euclidean liens. The general problem is to describe all metrics fromH which at each point possess the length indicatrix from a prescribed parametric class of convex figures. As a tool, a differential equation is proposed derived from the “triangular deficit principle” formulated in an earlier paper of R. V. Ambartzumian. The paper demonstrates that for the case where the length indicatrix is segmental this equation leads to a complete solution. Also, there is a partial result stating that in the case of Riemann metrics the orientation of the ellipsi should necessarily be a harmonic function.     

The K?hler-Einstein metric for some Hartogs domains over symmetric domains

We study the complete K?hler-Einstein metric of a Hartogs domain built on an irreducible bounded symmetric domain sW, using a power N ^μ of the generic norm of Ω. The generating function of the K?hler-Einstein metric satisfies a complex Monge-Ampère equation with a boundary condition. The domain is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by X ε [0, 1[. This allows us to reduce the Monge-Ampère equation to an ordinary differential equation with a limit condition. This equation can be explicitly solved for a special value μ₀ of μ. We work out the details for the two exceptional symmetric domains. The special value μ₀ seems also to be significant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.     

Global existence for defocusing cubic NLS and Gross–Pitaevskii equations in three dimensional exterior domains

We prove global wellposedness in the energy space of the defocusing cubic nonlinear Schrödinger and Gross–Pitaevskii equations on the exterior of a nontrapping domain in dimension 3. The main ingredient is a Strichartz estimate obtained combining a semi-classical Strichartz estimate [R. Anton, Strichartz inequalities for Lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, arxiv:math.AP/0512639, Bull. Soc. Math. France, submitted for publication] with a smoothing effect on exterior domains [N. Burq, P. Gérard, N. Tzvetkov, On nonlinear Schrödinger equations in exterior domains, Ann. I.H.P. (2004) 295–318].