Hamiltonian stationary Lagrangian surfaces in Hermitian symmetric spaces

This paper is the third of a series on Hamiltonian stationary Lagrangian surfaces. We present here the most general theory, valid for any Hermitian symmetric target space. Using well-chosen moving frame formalism, we show that the equations are equivalent to an integrable system, generalizing the C^2 subcase analyzed in the first article (arXiv:math.DG/0009202). This system shares many features with the harmonic map equation of surfaces into symmetric spaces, allowing us to develop a theory close to Dorfmeister, Pedit and Wu's, including for instance a Weierstrass-type representation. Notice that this article encompasses the article mentioned above, although much fewer details will be given on that particular flat case

A canonical structure on the tangent bundle of a pseudo- or para-K\"ahler manifold

It is a classical fact that the cotangent bundle T^* \M of a differentiable manifold \M enjoys a canonical symplectic form $\Omega^*$. If (\M,\j,g,\omega) is a pseudo-K\"ahler or para-K\"ahler $2n$-dimensional manifold, we prove that the tangent bundle T\M also enjoys a natural pseudo-K\"ahler or para-K\"ahler structure (\J,\G,\Omega), where $\Omega$ is the pull-back by $g$ of $\Omega^*$ and \G is a pseudo-Riemannian metric with neutral signature $(2n,2n)$. We investigate the curvature properties of the pair (\J,\G) and prove that: \G is scalar-flat, is not Einstein unless $g$ is flat, has nonpositive (resp.\ nonnegative) Ricci curvature if and only if $g$ has nonpositive (resp.\ nonnegative) Ricci curvature as well, and is locally conformally flat if and only if $n=1$ and $g$ has constant curvature, or $n>2$ and $g$ is flat. We also check that (i) the holomorphic sectional curvature of (\J,\G) is not constant unless $g$ is flat, and (ii) in $n=1$ case, that \G is never anti-self-dual, unless conformally flat.Comment: Clarified the statements on the cotangent bundle. Corrected various typo

Ricci curvature on polyhedral surfaces via optimal transportation

The problem of defining correctly geometric objects such as the curvature is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu & Yau, Jost & Liu have used and extended this notion for graphs giving estimates for the curvature and hence the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific but crucial case of polyhedral surfaces

Discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. A discrete Lawson correspondence

The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two discrete isothermic surfaces. We show that this correspondence is an isometry in the following sense: it preserves the metric coefficients introduced previously by Bobenko and Suris for isothermic nets. Exactly as in the smooth case, this is a correspondence between nets with the same Lax matrices, and the immersion formulas also coincide with the smooth case.Comment: 13 page

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure \J, a pseudo-metric \G with neutral signature and a symplectic structure \Om. We give a local classification of those surfaces of $T\Sigma$ which are both Lagrangian with respect to \Om and minimal with respect to \G. We first show that if $g$ is non-flat, the only such surfaces are affine normal bundles over geodesics. In the flat case there is, in contrast, a large set of Lagrangian minimal surfaces, which is described explicitly. As an application, we show that motions of surfaces in $\R^3$ or $\R^3_1$ induce Hamiltonian motions of their normal congruences, which are Lagrangian surfaces in $T\S^2$ or T \H^2 respectively. We relate the area of the congruence to a second-order functional $\mathcal{F}=\int \sqrt{H^2-K} dA$ on the original surface.Comment: 22 pages, typos corrected, results streamline

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

A rigidity theorem for Riemann's minimal surfaces

International audienceWe describe first the analytic structure of Riemann's examples of singly-periodic minimal surfaces; we also characterize them as extensions of minimal annuli bounded by parallel straight lines between parallel planes. We then prove their uniqueness as solutions of the perturbed problem of a punctured annulus, and we present standard methods for determining finite total curvature periodic minimal surfaces and solving the period problems

Bijective rigid motions of the 2D Cartesian grid

International audienceRigid motions are fundamental operations in image processing. While they are bijective and isometric in R^2, they lose these properties when digitized in Z^2. To investigate these defects, we first extend a combinatorial model of the local behavior of rigid motions on Z^2, initially proposed by Nouvel and Rémila for rotations on Z^2. This allows us to study bijective rigid motions on Z^2, and to propose two algorithms for verifying whether a given rigid motion restricted to a given finite subset of Z^2 is bijective