Pascal Romon

Hamiltonian Stationary Tori in the Complex Projective Plane

Hamiltonian stationary Lagrangian surfaces are Lagrangian surfaces in a four-dimensional Kahler manifold which are critical points of the area functional for Hamiltonian infinitesimal deformations. In this paper we analyze these surfaces in the complex projective plane: in a previous work we showed that they correspond locally to solutions to an integrable system, formulated as a zero curvature on a (twisted) loop group. Here we give an alternative formulation, using non-twisted loop groups and, as an application, we show in detail why Hamiltonian stationary Lagrangian tori are finite type solutions, and eventually describe the simplest of them: the homogeneous ones.

Ricci Curvature on Polyhedral Surfaces via Optimal Transportation

The problem of correctly defining 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 and Yau and Jost and 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.

Embedded minimal ends of finite type

We prove that the end of a complete embedded minimal surface in R^3 with infinite total curvature and finite type has an explicit Weierstrass representation that only depends on a holomorphic function that vanishes at the puncture. Reciprocally, any choice of such an analytic function gives rise to a properly embedded minimal end E provided that it solves the corresponding period problem. Furthermore, if the flux along the boundary vanishes, then the end is C^0-asymptotic to a Helicoid. We apply these results to proving that any complete embedded one-ended minimal surface of finite type and infinite total curvature is asymptotic to a Helicoid, and we characterize the Helicoid as the only simply connected complete embedded minimal surface of finite type in R ^3.

The periodic isoperimetric problem

Given a discrete group G of isornetries of R 3 , we study the G-isoperimetric problem, which consists of minimizing area (modulo G) among surfaces in R 3 which enclose a G-invariant region with a prescribed volume fraction. If G is a line group, we prove that solutions are either families of round spheres or right cylinders. In the doubly periodic case we prove that for most rank two lattices, solutions must be spheres, cylinders or planes. For the remaining rank two lattices we show, among other results, an isoperimetric inequality in terms of the topology of the isoperimetric surfaces. Finally, we study the case where G = Pm3m (the group of symmetries of the integer rank three lattice Z 3 ) and other crystallographic groups of cubic type. We prove that isoperimetric solutions must be spheres if the prescribed volume fraction is less than 1/6, and we give an isoperimetric inequality for G-invariant regions that, for instance, implies that the area (modulo Z 3 ) of a surface dividing the three space in two G-invariant regions with equal volume fractions, is at least 2.19 (the conjectured solution is the classical P Schwarz triply periodic minimal surface whose area is ∼ 2.34). Another consequence of this isoperimetric inequality is that Pm3m-symmetric surfaces (other than families of spheres) cannot be isoperimetric for the lattice group Z 3 .

The spectral data for Hamiltonian stationary Lagrangian tori in R^4

Abstract This article determines the spectral data, in the integrable systems sense, for all weakly conformally immersed Hamiltonian stationary Lagrangian in R 4 . This enables us to describe their moduli space and the locus of branch points of such an immersion. This is also an informative example in integrable systems geometry, since the group of ambient isometries acts non-trivially on the spectral data and the relevant energy functional (the area) need not be constant under deformations by higher flows.

Cyclic and ruled Lagrangian surfaces in Euclidean four space

We study those Lagrangian surfaces in complex Euclidean space which are foliated by circles or by straight lines. The former, which we call cyclic, come in three types, each one being described by means of, respectively, a planar curve, a Legendrian curve in the 3-sphere or a Legendrian curve in the anti-de Sitter 3-space. We describe ruled Lagrangian surfaces and characterize the cyclic and ruled Lagrangian surfaces which are solutions to the self-similar equation of the Mean Curvature Flow. Finally, we give a partial result in the case of Hamiltonian stationary cyclic surfaces.

Minimal Lagrangian surfaces in the tangent bundle of a Riemannian surface

Abstract Given an oriented Riemannian surface ( Σ , g ) , its tangent bundle T Σ enjoys a natural pseudo-Kahler structure, that is the combination of a complex structure J , a pseudo-metric G with neutral signature and a symplectic structure Ω . We give a local classification of those surfaces of T Σ which are both Lagrangian with respect to Ω 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 1 3 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 F = ∫ H 2 − K d A on the original surface.

Bijective Digitized Rigid Motions on Subsets of the Plane

Rigid motions in

Honeycomb geometry: Rigid motions on the hexagonal grid

\mathbb {R}^2