Weingarten Hypersurfaces and Flows

• Grant number: 23/06819-3
• Support Opportunities: Scholarships in Brazil - Doctorate
• Effective date (Start): January 01, 2024
• Effective date (End): July 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Alexandre Paiva Barreto
• Grantee: Rafael da Silva Belli
• Host Institution: Centro de Ciências Exatas e de Tecnologia (CCET). Universidade Federal de São Carlos (UFSCAR). São Carlos , SP, Brazil

Abstract

Darboux hypersurfaces are those obtained by moving a submanifold of codimension two by a one-parameter family of isometries of the ambient manifold. Important particular examples of Darboux hypersurfaces are the ruled hypersurfaces (in particular, cylindrical, conical and helicoidal hypersurfaces), translational and homothetic hypersurfaces, sphere-foliated hypersurfaces (in particular, rotational, tubular, cyclic, channel and osculating hypersurfaces) and Monge hypersurfaces.This project has two distinct parts, both of then having the Darboux hypersurfaces as a backdrop:The first part aims to study Weingarten hypersurfaces, that is, hypersurfaces whose mean curvatures verify a smooth relationship. Classic and deeply studied examples of these hypersurfaces are those with constant mean or Gauss-Kronecker (minimal hypersurfaces for example). Despite being a classic topic, many questions still remain open, even when the environment is the Euclidean Space. The objective of this part of the project is to develop techniques which allow to classify and/or produce relevant examples of nonlinear Darboux hypersurfaces immersed in Thurston model geometries or in their natural higher dimensional generalizations.Geometric flows defined by extrinsic curvatures have been one of the topics of Differential Geometry that have received the most attention from researchers in recent decades. The second part of the project is devoted to the study of Weingarten flows which are a natural generalization of mean curvature flow. More precisely, Weingarten flows are geometric flows whose value of the normal components of the time derivatives coincide with a smooth relationship between the mean curvatures of the hypersurfaces given by the flow. The aim of this part is to study the nonlinear Weingarten flows of Darboux hypersurfaces, to determine their existence and uniqueness, and to classify their translational and self-similar solitons.