Superfícies mínimas e uma família associada

Abstract

The aim of this monographis to explore the differential geometry of surfaces, with a special focus on minimal surfaces and their associated family, using a study of the geometric properties of a family associated with a minimal surface via isothermal parametrizations. For the development of this study, preliminary definitions are presented, where we discuss the quadratic forms of a regular surface, known as the first and second fundamental forms, the Christoffel symbols, and the compatibility equations (Gauss and Mainardi-Codazzi). Next, we develop the main objects of this study, which are the minimal surfaces. Finally, we prove that given a minimal surface S, which is not locally planar, there exists a phenomenon revealing the existence of a important family of surfaces associated with S, with the property that each surface in the family is a minimal surface, all of them have the same first fundamental form, and they are pairwise non-congruent. Lastly, we observe that the catenoid and the helicoid belong to the same family.