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- Frédéric Hélein, Pascal Romon
- 1999

We study Hamiltonian stationary Lagrangian surfaces in C 2 , i.e. Lagrangian surfaces in C 2 which are stationary points of the area functional under smooth Hamiltonian variations. Using loop groups, we propose a formulation of the equation as a completely integrable system. We construct a Weierstrass type representation and produce all tori through either… (More)

- Frédéric Hélein, Pascal Romon
- 2000

In this paper we revisit and generalize some aspects of the Weierstrass construction for Hamiltonian stationary Lagrangian surfaces which was given in HR]. We are interested here in Lagrangian surfaces in R 4 ' C 2 , equipped with the standard symplectic form, without the assumption of being Hamiltonian stationary, a priori. In order to build a Weierstrass… (More)

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,… (More)

- TORI IN R, Ian McIntosh, Pascal Romon
- 2010

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… (More)

- K. LESCHKE, P. ROMON
- 2008

The multiplier spectral curve of a conformal torus f : T 2 → S 4 in the 4– sphere is essentially [3] given by all Darboux transforms of f. In the particular case when the conformal immersion is a Hamiltonian stationary torus f : T 2 → R 4 in Euclidean 4–space, the left normal N : M → S 2 of f is harmonic, hence we can associate a second Riemann surface: the… (More)

- Pascal Romon
- 2004

We study Lagrangian submanifolds foliated by (n − 1)-spheres in R 2n for n ≥ 3. We give a parametrization valid for such submanifolds, and refine that description when the submanifold is special Lagrangian, self-similar or Hamiltonian stationary. In all these cases, the submanifold is centered, i.e. invariant under the action of SO(n). It suffices then to… (More)

- Henri Anciaux, Brendan Guilfoyle, Pascal Romon
- 2008

Given an oriented Riemannian surface (Σ, g), its tangent bundle T Σ enjoys a natural pseudo-Kähler 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… (More)

- Pascal Romon
- 2007

We describe here a general method for nding symmetries of minimal surfaces in R 3 , namely transformations sending a minimal immersion to another minimal immersion. More speciically we will be looking for innnitesimal symmetries, i.e. vector elds tangent to a Lie group acting on the set of minimal surfaces. Using NNther's theorem, we derive conserved… (More)

- Henri Anciaux, Pascal Romon
- 2007

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… (More)