Pascal Romon

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We study Hamiltonian stationary Lagrangian surfaces in C, i.e. Lagrangian surfaces in C 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 the(More)
We derive a Weierstrass-type formula for conformal Lagrangian immersions in Euclidean 4-space, and show that the data satisfies an equation similar to Dirac equation with complex potential. Alternatively this representation has a simple formulation using quaternions. We apply it to the Hamiltonian stationary case and construct all possible tori, thus(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 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 in Euclidean 4–space, the left normal N : M → S of f is harmonic, hence we can associate a second Riemann surface: the(More)
Euclidean rotations in Rn are bijective and isometric maps. Nevertheless, they lose these properties when digitized in Zn. For n = 2, the subset of bijective digitized rotations has been described explicitly by Nouvel and Rémila and more recently by Roussillon and Cœurjolly. In the case of 3D digitized rotations, the same characterization has remained an(More)
We present, in details, a generic tool to estimate differential geometric quantities on digital shapes, which are subsets of Zd . This tool, called digital integral invariant, simply places a ball at the point of interest, and then examines the intersection of this ball with input data to infer local geometric information. Just counting the number of input(More)
Hamiltonian stationary Lagrangian submanifolds are solutions of a natural and important variational problem in Kähler geometry. In the particular case of surfaces in Euclidean 4-space, it has recently been proved that the Euler–Lagrange equation is a completely integrable system, which theory allows us to describe all such tori. This article determines the(More)
We describe here a general method for finding symmetries of minimal surfaces in R3, namely transformations sending a minimal immersion to another minimal immersion. More specifically we will be looking for infinitesimal symmetries, i.e. vector fields tangent to a Lie group acting on the set of minimal surfaces. Using Nœther’s theorem, we derive conserved(More)