Joachimsthal surfaces with nonzero constant Gaussian curvature

• Grant number: 21/10181-9
• Support Opportunities: Scholarships abroad - Research Internship - Post-doctor
• Start date: March 31, 2022
• End date: June 29, 2022
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Ruy Tojeiro de Figueiredo Junior
• Grantee: Marcos Paulo Tassi
• Supervisor: José Antonio Gálvez López
• Host Institution: Instituto de Ciências Matemáticas e de Computação (ICMC). Universidade de São Paulo (USP). São Carlos , SP, Brazil
• Institution abroad: Universidad de Granada (UGR), Spain
• Associated to the scholarship: 20/03431-6 - Uniqueness of immersed spheres in three-dimensional Riemannian manifolds and Enneper-type hypersurfaces, BP.PD

Abstract

The aim of this project is to investigate Joachimsthal hypersurfaces in R3 with relevant geometric properties, such as constant Gaussian or constant mean curvatures, Weingarten surfaces, etc., in the light of the explicit description of such hypersurfaces in terms of certain differentiable functions of one variable and certain curves in the hyperbolic plane whose coordinate functions satisfy certain second order ODEs, recently obtained by the fellowship holder and his supervisor based on the results obtained by S. Chión and R. Tojeiro in 2021 (see https://arxiv.org/abs/2105.13842). We intend to develop a qualitative study of such ODEs and to investigate global aspects of Joachimsthal surfaces, such as the description of compact examples, in particular those of genus zeroand one. (AU)