Resumen de Problems about Mean Curvature in R^{n+1}

Eddygledson Souza Gama

• 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 assuming that the base space M has a particular structure. The third and last chapter of this thesis is devoted to proving a result of the characterization of translating 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}.