Enneper surfaces

• Grant number: 21/03304-7
• Support Opportunities: Scholarships in Brazil - Master
• Start date: October 01, 2021
• End date: February 28, 2023
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ruy Tojeiro de Figueiredo Junior
• Grantee: Alan Sousa França
• Host 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)

Academic Publications

(References retrieved automatically from State of São Paulo Research Institutions)

FRANÇA, Alan Sousa. Surfaces of Enneper type. 2023. Master's Dissertation - Universidade de São Paulo (USP). Instituto de Ciências Matemáticas e de Computação (ICMC/SB) São Carlos.