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

A smooth flat Riemannian manifold is called an exceptional domain if it admits positive harmonic functions having vanishing Dirichlet boundary data and constant (nonzero) Neumann data. In analogy with minimal surfaces, a representation formula derived applied to the classification of domains. Some interesting open problems are proposed along way.

The wavefront set provides a precise description of the singularities distribution. Because its ability to control product distributions, was key element recent progress in renormalized quantum field theory curved spacetime, gravity, discussion time machines or energy inequalitites. However, is somewhat subtle concept whose standard definition not easy grasp. This paper step by introduction set, with examples and motivation. Many different definitions new interpretations are presented. Some...

We propose a characterisation of Willmore immersions inspired from the works R. Bryant on surfaces and J. Dorfmeister, F. Pedit, H.-Y. Wu harmonic maps between surface compact homogeneous manifold using moving frames loop groups.We use that formulation in order to construct Weierstrass type representation all conformal terms closed one-forms.Let R 3 be Euclidean space let us consider set D compact, oriented without boundary which are immersed (the immersion being class C for k > 4).For S G...

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 with complex potential.Alternatively this representation has simple formulation using quaternions.We apply it Hamiltonian stationary case construct all possible tori, thus obtaining first approach moduli space terms of algebraic-geometric problem on plane.We also classify Klein bottles they self-intersect.

We study Hamiltonian stationary Lagrangian surfaces in C 2 , i.e., which are points of the area functional under smooth variations.Using loop groups, we propose a formulation equation as completely integrable system.We construct Weierstrass type representation and produce all tori through either systems machinery or more direct arguments. Introduction.This paper addresses oriented symplectic Euclidean vector space dimension 4, using techniques systems.The ambient may be seen with, complex...

The main purpose in the present paper is to build a Hamiltonian theory for fields which consistent with principles of relativity.For this we consider detailed geometric pictures Lepage theories spirit Dedecker and try stress out interplay between Lepage-Dedecker (LP) description (more usual) De Donder-Weyl (DDW) one.One points fact that Legendre transform DDW approach replaced by correspondence LP (this behaves differently: ignoring singularities whenever Lagrangian degenerate).

This papers is concerned with multisymplectic formalisms which are the frameworks for Hamiltonian theories fields theory.Our main purpose to study observable (n -1)-forms allows one construct functionals on set of solutions Hamilton equations by integration.We develop here two different points view: generalizing law {p, q} = 1 or dF/dt {H, F }.This leads possible definitions; we explore relationships and differences between these concepts.We show that -in contrast de Donder-Weyl theory -the...

We analyze here Hamiltonian stationary surfaces in the complex projective plane as (local) solutions to an integrable system, formulated a zero curvature on loop group. As application, we show details why such tori are finite type solutions, and eventually describe simplest of them: homogeneous ones.

The pull-back, push-forward and multiplication of smooth functions can be extended to distributions if their wave front sets satisfy some conditions. Thus, it is natural investigate the topological properties these operations between spaces ${\mathc

The formulation of a relativistic dynamical problem as system Hamilton equations by respecting the principles Relativity is delicate task, because in their classical form require use time coordinate, which course contradicts Relativity. Two interesting solutions have been proposed during last century: covariant phase space and multisymplectic formalism. These two approaches were inspired at beginning different points view. However, shown works Kijowski-Szczyrba, Forger-Romero Vitagliano,...