Contribuições à teoria das superfícies de curvatura média constante

Abstract

n 1985, R. Bryant ([Br]) carried out a study in which he sought to determine which spatial classes of spatial forms of dimension 3 allowed local representation in terms of holomorphic data, as had been done by Enneper and Weierstrass in minimum surfaces. As a result of their investigations, we conclude that only a new case appeared, namely surfaces with constant mean curvature equal to 1 (CMC 1) in the hyperbolic space with sectional curvature equal to -1. Classically, it is known that surfaces were locally isometric to surfaces through Darboux-Lawson's correspondence, but Bryant went further and showed that from the global point of view the analogy was held in the sense that CMC 1 surfaces also allowed an Enneper-Weierstrass representation. In particular, he showed that some classic minimal surfaces, such as the catenode and the surface of Enneper, had hyperbolic correspondences, which he called "raw". Bryant's representation subsequently allowed the construction of numerous global examples, notably by W. Rossman, M. Umehara and K. Yamada. In particular, in ([RUY]), these authors showed that, starting from a minimal surface satisfying a set of natural geometric conditions, it was possible to construct a family to a parameter of CMC 1 surfaces and thus provided many other examples of surfaces cousins. In view of this, it is natural to seek to ascertain whether such construction can be carried out starting from the aforementioned Costa-Hoffmann-Meeks-Karcher surfaces.