Frédéric Hélein and

Some examples of ‘second order elliptic integrable systems associated to a 4-symmetric space’

From cmc surfaces to Hamiltonian stationary Lagrangian surfaces

Minimal surfaces and surfaces with constant mean curvature (cmc) have fascinated differential geometers for over two centuries. Indeed these surfaces are solutions to variational problems whose formulation is elegant, modelling physical situations quite simple to experiment using soap films and bubbles; however their richness has not been exhausted yet. Advances in the comprehension of these su...

FOUR LAMBDA STORIES , AN INTRODUCTION TO COMPLETELY INTEGRABLE SYSTEMS by Frédéric

— Among all non-linear differential equations arising in Physics or in geometry, completely integrable systems are exceptional cases, at the concurrence of miraculous symmetry properties. This text proposes an introduction to this subject, through a list of examples (the sinh-Gordon, Toda, Korteweg-de Vries equations, the harmonic maps, the anti-self-dual connections on the four-dimensional spa...

Curved space-times by crystallization of liquid fiber bundles

Motivated by the search for a Hamiltonian formulation of Einstein equations of gravity which depends in a minimal way on choices of coordinates, nor on a choice of gauge, we develop a multisymplectic formulation on the total space of the principal bundle of orthonormal frames on the 4-dimensional space-time. This leads quite naturally to a new theory which takes place on 10-dimensional manifold...

Hamiltonian formalism with several variables and quantum field theory: Part I

We discuss in this paper the canonical structure of classical field theory in finite dimensions within the pataplectic hamiltonian formulation, where we put forward the role of Legendre correspondance. We define the Poisson p-brackets and ωbrackets which are the analogues of the Poisson bracket on forms. We formulate the equations of motion of forms in terms of p-brackets and ω-brackets with th...

A Note on Some Overdetermined Elliptic Problem

Ω 7−→ λ1(Ω) , under the volume constraint Vol(Ω) = α (where α ∈ (0,Vol(M)) is fixed) are called extremal domains. Smooth extremal domains are characterized by the property that the eigenfunctions associated to the first eigenvalue of the Laplace-Beltrami operator have constant Neumann boundary data [2]. In other words, a smooth domain is extremal if and only if there exists a positive function ...