Propriedades características das hiperesferas euclidianas
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Data
2008-06-06
Autores
Lozório, Weslley Marinho
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Universidade Federal do Espírito Santo
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The study of hypersurfaces of Euclidean spaces which have a constant elementary symmetric function is a classical topic in Differential Geometry. In this topic the more simple geometric problem is to characterize the compact hypersurfaces and the prototypical result was obtained by H. Liebmann in 1899: the round spheres are the only compact surfaces in the three dimensional Euclidean space that have constant Gaussian curvature. In 1956 A.D. Alexandrov obtained a remarkable characterization of the Euclidean round hyperspheres: they are the only compact hypersurfaces of m-dimensional Euclidean space (m ¸ 3) that have constant mean curvature. The ideas used by Alexandrov became well-know as Alexandrovs reflection method and were used in several other problems. In 1977, R.C. Reilly presented a new proof of Alexandrovs theorem, the Reillys method, which also become fundamental tool in this topic. In fact, A. Ros in 1987, using the Reillys method, obtained a new extension of the Alexandrovs theorem characterizing the round hyperspheres as the only compact hypersurfaces of the m-dimensional Euclidean space that have a constant elementary symmetric function of the principal curvatures. This result implies, in particular, the Liebmanns theorem. In 1988, N. Korevaar presented a new proof of the Ross theorem, using the Alexandrov reflection method. The main goal of this Master thesis is to present proofs by Alexandrov, Reilly, Ros, and Korevaar of some theorems that characterizes the Euclidean round hyperspheres
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LOZÓRIO, Weslley Marinho. Propriedades Características das Hiperesferas Euclidianas. 2008. 72 f. Dissertação (Mestrado em Matemática) - Programa de Pós-Graduação em Matemática, Universidade Federal do Espírito Santo, Vitória, 2008.