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Deformation and construction of minimal surfaces

Abstract : This thesis is devoted to the construction of numerous examples of minimal surfaces (or hypersurfaces) in the 3-Euclidean space, R^n x R with n≻2 or in the homogeneous space S² x R . We prove the existence of minimal surfaces in R³ as close as we want of a convex polygon. We prove the existence of minimal hypersurfaces in R^n x R, n≻2, whose have Riemann's type. These ones could be considered as a family of horizontal hyperplanes (the ends) which are linked to each other by pieces of deformed catenoids (the necks). We provide a general result in the case simply-periodic together with the case of a finite number of hyperplanar ends. Our construction lies on some conditions associates with the forces that characterize the different configurations. We end with giving some examples ; in particular, we exhibit the existence of vertical Wei example that does not exists in the 3-dimensional case. We also prove the existence of the analogous of the Wei example in S² x R. The surface is such that two spherical ends are linked by 1 neck and 2 necks alternatively. Here again, we highlight the role that the forces play in the construction. Moreover, like in the previous chapter, the method lies on a gluing process. We give an accurate description of the catenoid and the Riemann's minimal example in S² x R. Finally, we demonstrate the existence of Scherk type hypersurfaces in R^n x R when n≻2
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Submitted on : Tuesday, March 19, 2013 - 5:02:16 PM
Last modification on : Tuesday, November 3, 2020 - 10:02:23 AM


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Antoine Coutant. Deformation and construction of minimal surfaces. General Mathematics [math.GM]. Université Paris-Est, 2012. English. ⟨NNT : 2012PEST1069⟩. ⟨tel-00802379⟩



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