Problèmes variationnels liés à l'aire

Abstract : The main focus of my work is the classification and rigidity properties of the critical points of the area functional -- minimal surfaces and the like -- for surfaces in Euclidean or more general homogeneous spaces. The setting is Riemannian or Hermitian. I have tried to understand and describe the structure of the PDE corresponding to the geometric variational problem, and of its solutions. Using conformal parametrizations, I have characterized those solutions that satisfy certain geometric or topological requirements, such as: embeddedness, period closing condition (genus 1) or isoperimetry.

In the first part of this work, I consider classical minimal surfaces in three dimensional Euclidean space, whose analytical structure is given by Weierstrass representation formula. Using this formula, one reduces a topologically or geometrically constrained problem (e.g. number of ends, finite total curvature, simple periodicity) to a complex analysis problem on a Riemann surface. I obtain thus a rigidity theorem for Riemann minimal staircase. However my most important results bear on the behavior of embedded minimal ends of infinite total curvature and finite type. I have shown indeed that the embeddedness condition alone limits considerably the Weierstrass data, practically forcing the end to be asymptotic to the helicoid. This result plays a part in the recent proof by Meeks and Rosenberg of the uniqueness of the helicoid as properly embedded simply connected minimal surface.

In the second part, I display various results on the isoperimetric problem in flat periodic spaces in three dimensions. This is still today an open problem involving constant mean curvature surfaces. I have in particular worked on the sphere-cylinder-plane conjecture in three dimensional tori and obtained sharp inequalities for the cases where the conjecture holds. I have also proven that surfaces with too many symmetries in the cube cannot be isoperimetric, but for spheres. Finally a numerical study hints at the reason why this problem remains so difficult.

In the third part are my results on Hamiltonian stationary Lagrangian surfaces, in Euclidean four space and in Hermitian symmetric spaces as well. I show how the associated PDE is an integrable system (as in the CMC case). From that various applications are proven, such as construction of finite type tori, Weierstrass-type potential representation (using loop groups). This approach is further refined in the Euclidean case into a spinor representation that actually solves even the period problem (for tori). A generalization to higher dimensions is sketched.
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https://tel.archives-ouvertes.fr/tel-00008760
Contributor : Pascal Romon <>
Submitted on : Friday, March 11, 2005 - 5:15:28 PM
Last modification on : Wednesday, September 4, 2019 - 1:52:03 PM
Long-term archiving on : Friday, April 2, 2010 - 9:41:30 PM

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  • HAL Id : tel-00008760, version 1

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Pascal Romon. Problèmes variationnels liés à l'aire. Mathématiques [math]. Université de Marne la Vallée, 2004. ⟨tel-00008760⟩

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