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Construction de surfaces minimales par résolution du problème de Dirichlet

Abstract : This Ph.D. thesis deals with the theory of minimal surfaces. In 2001, C. Cosin and A. Ros show that, if a polygon bounds an immersed disk, this polygon is the flux polygon of a symmetric Alexandrov-embedded r-noid with genus 0. Their proof is based on the study of the space of these minimal surfaces. Our work gives a more constructive proof of their result. Our method is based on the solving of the Dirichlet problem for the minimal surface equation. To this end, we study the convergence of sequences of the equation's solutions. We define the lines of divergence which are the points where the gradients' sequence is unbounded. The study of these lines allows us to conclude on the sequences' convergence. The r-noids are then build as the conjugate surfaces to the graphs of Dirichlet problem's solutions on domains fixed by the polygons. In a second part, we show that, under the "immersed disk bounding" hypothesis, a polygon is also the flux polygon of a genus 1 symmetric Alexandrov-embedded r-noid. The proof is based on the same ideas as the first result, however it requires the solving of a period problem. This solving uses the study of the limit behaviour of some minimal surfaces sequences.
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Contributor : Laurent Mazet <>
Submitted on : Friday, December 17, 2004 - 10:57:10 AM
Last modification on : Friday, January 10, 2020 - 9:08:06 PM
Long-term archiving on: : Thursday, September 13, 2012 - 1:45:38 PM


  • HAL Id : tel-00007780, version 1



Laurent Mazet. Construction de surfaces minimales par résolution du problème de Dirichlet. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2004. Français. ⟨tel-00007780⟩



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