DSpace Coleção:http://www.repositorio.ufc.br/handle/riufc/652019-10-16T06:53:29Z2019-10-16T06:53:29ZProblems about mean curvatureGama, Eddygledson Souzahttp://www.repositorio.ufc.br/handle/riufc/450032019-08-22T13:25:59Z2019-07-25T00:00:00ZTítulo: Problems about mean curvature
Autor(es): Gama, Eddygledson Souza
Abstract: This thesis is divided into three chapters. In the ﬁrst chapter, it is done a brief introduction of the main tools necessary for the development of this work. In turn, in the second chapter it develops the Jenkins-Serrin theory for vertical and horizontal cases. Regarding the vertical case, it only proves the existence of the solution of Jenkins-Serrin problem for the type I, when M is rotationally symmetric and has non-positive sectional curvatures.However, with respect to the horizontal case, the existence and the uniqueness is proved
in a general way, namely a.ssuming that the base space M has a particular structure. The ing solitons in R
n+1 . More precisely, it is proved that the unique examples C 1 −asymptotic to two half-hyperplanes outside a cylinder are the hyperplanes parallel to e n+1 and the elements of the family associated with the tilted grim reaper cylinder in R n+1 .2019-07-25T00:00:00ZDifferential operators penalized by geometric potentials.Souza, Leo Ivo da Silvahttp://www.repositorio.ufc.br/handle/riufc/444892019-08-27T12:31:42Z2018-08-21T00:00:00ZTítulo: Differential operators penalized by geometric potentials.
Autor(es): Souza, Leo Ivo da Silva
Abstract: This paper is presented in two parts. In the ﬁrst part, we establish the non-positivity of the second eigenvalue of the Schrödinger operator −div P r ∇ · − W 2r on a closed hypersurface Σ n of Rn+1 , where W r is a power of the (r + 1)-th mean curvature of Σ n which we will ask to be positive. If this eigenvalue is null, we will have a characterization of the sphere. This theorem generalizes the result of Harrell and Loss proved to the Laplace-Beltrame operator penalized by the square of the mean curvature. In the second part, we established the non-positivity of the second auto-value of the Schödinger operator − d2ds2 − (√F) −2CF(κ), in a closed curve of the plane with length 2π, F ∈ C 1 ( R ) and κ is the curvature of the curve. If this eigenvalue is null, we will have a characterization of the circle, which generalizes partially the result of Harrell and Loss proved to the one-dimensional Laplace operator penalized by the square of the curvature in curves of the plane.2018-08-21T00:00:00ZPrimeiro autovalor do operador de Laplace penalizado pela curvatura média e o funcional de Willmore.Vieira, Francisca Damianahttp://www.repositorio.ufc.br/handle/riufc/435042019-07-18T16:08:16Z2019-06-18T00:00:00ZTítulo: Primeiro autovalor do operador de Laplace penalizado pela curvatura média e o funcional de Willmore.
Autor(es): Vieira, Francisca Damiana
Abstract: In this work, we will prove some results for the ﬁrst eigenvalue of a linear diﬀerential Schrödinger operator L = −Δ − (1/n)H*2, deﬁned on closed hypersurfaces with the same volume of the sphere and immersed in Rn+1 , where −Δ is the Laplace-Beltrami operator and H = Pnj=1kj , with kj the hypersurface principal curvatures. Under these conditions, we will show a local generalization for the classical result of the Willmore functional for the Euclidean sphere. As a consequence, we will prove that the ﬁrst eigenvalue of this operator in the Euclidean sphere is a local maximum and this result is a global one in the closed hypersurface space of R3 and genus zero.2019-06-18T00:00:00ZCampos conformes e métricas críticas em variedades compactas com bordo.Viana, Emanuel Mendonçahttp://www.repositorio.ufc.br/handle/riufc/434042019-07-05T11:44:17Z2019-06-27T00:00:00ZTítulo: Campos conformes e métricas críticas em variedades compactas com bordo.
Autor(es): Viana, Emanuel Mendonça
Abstract: This work is divided into two parts and it aims to study conformal vector ﬁelds and critical metrics on compact manifold with smooth boundary. The ﬁrst of these parts is related to compact Riemannian manifold (M n , g) with smooth boundary under the existence of nontrivial conformal gradient vector ﬁeld. With appropriate controls on the Ricci’s curvature, we show that M is isometric to a hemisphere of the sphere, where we use the stiﬀness results of Reilly (1977 e 1980). Next, considering the case in which the manifold is Einstein with the existence of nonzero conformal gradient vector ﬁeld, we prove that its scalar curvature is positive and it must be isometric to a hemisphere of Sn . Finally, we conclude that part by showing that a suitable control on the energy of a conformal vector ﬁeld implies that M is isometric to a hemisphere S+n. In the second part, we study compact Riemannian manifolds (M n , g) that admit a non-constant solution to the system of equations −Δf g + Hessf − fRic = µRic + λg, where Ric is the Ricci tensor of g where as µ and λ are two real parameters. More precisely, under assumption that (M n , g) has zero radial Weyl curvature, this means that the interior product of ∇f with the Weyl tensor W is zero, we shall provide the complete classiﬁcation for the following structures: positive static triples, critical metrics of volume functional and critical metrics of the total scalar curvature functional.2019-06-27T00:00:00Z