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Enneper surfaces

Grant number: 21/03304-7
Support type:Scholarships in Brazil - Master
Effective date (Start): October 01, 2021
Effective date (End): March 31, 2023
Field of knowledge:Physical Sciences and Mathematics - Mathematics - Geometry and Topology
Principal researcher:Ruy Tojeiro de Figueiredo Junior
Grantee:Alan Sousa França
Home Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
Associated research grant:16/23746-6 - Algebraic, topological and analytical techniques in differential geometry and geometric analysis, AP.TEM

Abstract

An umbilic-free surface in Euclidean space R^3 whose curvature lines correspondent to one of its principal curvatures are contained in spheres or planes is called an Enneper surface. In this project we present a parametrization of Enneper surfaces with one family of planar curvature lines in terms of Ribaucour partial tubes, introduced in recent work by S. Chión and R. Tojeiro, and describe how an arbitrary Enneper surface can be constructed from an Enneper surface with one family of planar curvature lines. We also present a new description, obtained in S. Chión and R. Tojeiro, of the special class of Enneper surfaces with the property that the spheres that contain the curvature lines correspondent to one of its principal curvatures are all centered on a common straight line, called Joachimsthal surfaces, based on the conformal diffeomorfism of H^2 X R onto R^3 - R. We provide a proof of the classical fact that any Enneper surface with non constant Gauss curvature is a Joachimsthal surface, and the explicit description of such surfaces given recently by M. Tassi and R. Tojeiro. Finally, we discuss the description of minimal and constant mean curvature Enneper surfaces in R^3, as well as some related results in hyperbolic space H^3. (AU)

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