Pascal Romon

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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)
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)
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)
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)
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)