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Sobre hipersuperfÃcies r-mÃnimas com fins planares no espaÃo euclidiano. / On hipersurface r-minims with ends to glide in the Euclidean spaceJuscelino Pereira da Silva 21 September 2007 (has links)
CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Uma hipersuperficie sigma està contido Rn+1 à r-mÃnima se sua (r + 1)-curvatura (a (r + 1)-Ãsima funÃÃo simÃtrica elementar de suas curvaturas principais) à identicamente nula. Se n > 2(r + 1)mostramos que a hipersuperfÃcie r-mÃnima rotacionalmente invariante en Rn+1, a saber, o n-catenÃide, descrito em [HL1], à nÃo-degenerado no sentido que nÃo possui campos de Jacobi que decaem suficientemente rÃpido no infinito. Combinando isto com a teoria de deformaÃÃo
em espaÃos de Holder com peso desenvolvida por Mazzeo, Pacard, Pollack, Uhlenbeck e outros,obtemos novos resultados sobre a estrutura de hipersuperfÃcies r-mÃnimas com fins planares.
Por exemplo, mostramos que o espaÃo moduli Mr,k de hipersuperfÃcies completas r-mÃnimas elÃpticas no espaÃo euclidiano Rn+1, n > 2(r + 1), com k > 2 fins planares, tem a estrutura de variedade analÃtica de dimensÃo formal k(n + 1), que à realizada na vizinhanÃa de umelemento nÃo-degenerado. Mais ainda, produzimos novos exemplos de famÃlias de dimensÃo infinita de hipersuperfÃcies r-mÃnimas obtidas por perturbaÃÃes de catenÃides truncados. / A hypersurface sigma Rn+1 is r-minimal if its (r + 1)th-curvature (the (r + 1)th elementary symmetric function of its principal curvatures) vanishes identically. If n > 2(r + 1) we show that the rotationally invariant r-minimal hypersurfaces in Rn+1 (catenoids) first described in
[HL1] are nondegenerate in the sense that they do not carry Jacobi fields which decay rapidly enough at infinity. Combining this with the deformation theory in weighted Holder spaces developed by Kusner, Mazzeo, Pacard, Pollack, Uhlenbeck and others, we obtain new results
on the structure of r-minimal hypersurfaces with ends of planar type. For example, we show that the moduli space Mr,k of complete r-minimal hypersurfaces in Euclidean space Rn+1,n > 2(r+1), with k > 2 ends of planar type has the structure of an analytic manifold of virtual dimension k(n+1), which is attained in a neighborhood of a nondegenerate element. Also, we produce new infinite dimensional families of examples of r-minimal hypersurfaces obtained by perturbing noncompact portions of the catenoids. These seem to be the first known families
of examples of noncompact elliptic r-minimal hypersurfaces without symmetries
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