DSpace Coleção:
http://www.repositorio.ufc.br/handle/riufc/65
2021-10-21T09:19:30ZHipersuperfícies r-mínimas no espaço euclidiano
http://www.repositorio.ufc.br/handle/riufc/60911
Título: Hipersuperfícies r-mínimas no espaço euclidiano
Autor(es): Sousa, Paulo Alexandre Araújo
Abstract: n the first part (chapters 2, 3 and 4) of this Thesis we will study the hypersurfaces of R p+q+2 that are r-minimum (Sr = 0) and invariant by the canonical action of the group O(p + 1) × O(q + 1). We will get a rating
complete of all hypersurfaces of R p+q+2 which are O(p + 1) × O(q + 1)-invariant and have the r-'th null mean curvature (2 ≤ r ≤ min{p, q} ), analyzing whether such hypersurfaces are complete, embedded and (r − 1)-stable. With this we will obtain the following existence result: “Let p, q, r ∈ N be such that p + q ≥ r + 5 and 2 ≤ r ≤ min{p, q}, then there is a hypersurface Mp+q+1 ⊂ R p+q+2 complete, layered, with r-´th null mean curvature which is globally (r − 1)-stable”. In chapter 5 we will study the cones C(M) ⊂ R n+1 r-minimum, whose base Mn−1 ⊂ S n is a compact hypersurface such that Sr = 0 and Sr+1 is a non-zero constant. We will prove that: “If r + 2 ≤ n ≤ r + 5, then there is 0 < ε < 1 such that the trunk of
cone C(M)ε is not (r − 1)-stable”. Furthermore, we will construct a Clifford Torus with Sr = 0 and Sr+1 6 = 0 to show that this result is not valid when n ≥ r + 6.2007-01-01T00:00:00ZHipersuperfícies r-mínimas com dois fins regulares
http://www.repositorio.ufc.br/handle/riufc/60908
Título: Hipersuperfícies r-mínimas com dois fins regulares
Autor(es): Sousa, Antonio Fernando Pereira de
Abstract: Let Mn be a r-minimal hypersurface in R n+1, i.e., suppose M has curvature S r+1 identically zero. M is said regular if out of any compact M is the disjunct union of a finite number of ends, each regular, i.e., with the same assymptotic behavior that a rotational hypersurface. It is shown that embedded, elliptic rminimal hypersurfaces in Euclidean space Rn+1,3/2(r + 1) ≤ n < 2(r + 1), with two ends, both regular, are catenoids (i.e. rotational hypersurfaces). This extends previous results by Schoen [7] and Hounie-Leite [3].2008-01-01T00:00:00ZSobre hipersuperficies r-minimas com fins planares no espaço euclidiano.
http://www.repositorio.ufc.br/handle/riufc/60867
Título: Sobre hipersuperficies r-minimas com fins planares no espaço euclidiano.
Autor(es): Silva, Juscelino Pereira
Abstract: A hypersurface Σ ⊂ 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 showthat the rotationally invariant r-minimal
hypersurfaces in R n+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 H¨older 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 R n+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.2007-01-01T00:00:00ZFuncionais paramétricos elípticos em variedades riemannianas
http://www.repositorio.ufc.br/handle/riufc/60855
Título: Funcionais paramétricos elípticos em variedades riemannianas
Autor(es): Melo, Marcelo Ferreira de
Abstract: It is stated that critical points of a parametric elliptic functional in a Riemannian manifold are hypersurfaces with prescrebed anisotropic mean curvature. We prove that the anisotropic Gauss map of surfaces immersed in Euclidean space with constant anisotropic mean curvature is a harmonic map. In the case of rotatioally invariat functionals in some homogeneous three-dimensional ambients, we present a abridged version of a existence
result for constant anisotropic mean curvature surfaces as cylinders, spheres, tori and annuli corresponding to the anisotropic analogs of onduloids and nodoids. In the Euclidean case M¯ = R3, examples of stable critical points are provided by theWulff shapes associated to functional F. Paralleling the case of constant curvature
mean spheres, a characterization of Wulff shapes is provided, which answers affirmatively a question posed by M. Koiso and B. Parmer in [13].2009-01-01T00:00:00Z