Closed minimal hypersurfaces in noncompact symmetric spaces

• Grant number: 07/08513-6
• Support Opportunities: Scholarships in Brazil - Doctorate
• Effective date (Start): April 01, 2008
• Effective date (End): March 31, 2012
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Claudio Gorodski
• Grantee: Jaime Leonardo Orjuela Chamorro
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 07/03192-7 - Submanifold geometry and Morse theory in finite and infinite dimensions, AP.TEM

Abstract

The classical local invariants of a submanifold of a space form are the first fundamental form, the shape operators and the induced normal connection. They determine the submanifold up to an isometry of the ambient space. Isoparametric submanifolds of space forms are submanifolds with very simple invariants. Namely, a submanifold with constant principal curvatures is a submanifold whose principal curvatures with respect to a parallel normal vector field along a piecewice smooth curve are constant. An isoparametric submanifold of a space form is a submanifold with constant principal curvatures and flat normal bundle. Suggested questions for investigation.1. A compact isoparametric hypersurface of a hypersphere of a Euclidean space can be described as a level set of an isoparametric polinomial satisfying the so called Cartan-Münzner equations. Study this system from the viewpoint of partial differential equations. For example, study the linearization of the system in order to contribute to the classification problem that has not been completely solved yet.2. The Cartan-Müzner equations have been generalizated to isoparametric submanifolds of more general ambient spaces. In particular, the previous problem can be considered in the context of equifocal submanifolds of symmetric spaces of compact type. 3. Uses the method of moving frames to study isoparametric hypersurfaces of spheres with all multiplicities equal to one.a. Give an alternative proof of the homogeneity in the case of g=6 different principal curvatures. b. Give an alternative proof that the number g of different principal curvatures must be equal to1, 2, 3, 4 or 6. 4. Study isoparametric hypersurfaces of Hilbert spaces with all multiplicites equal to one (use the method of moving frames).5. Study equifocal submanifolds of symmetric spaces of compact type with all multiplicites equal to one (use the relation betwen isoparametric submanifolds of Hibert spaces and equifocal submanifolds of symmetric spaces). (AU)

Scientific publications

(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)

ORJUELA CHAMORRO, JAIME LEONARDO. Complete minimal hypersurfaces in quaternionic hyperbolic space. Geometriae Dedicata, v. 172, n. 1, p. 47-67, OCT 2014. (07/08513-6)

Academic Publications

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

CHAMORRO, Jaime Leonardo Orjuela. Complete minimal hipersurfaces in symmetric spaces. 2012. Doctoral Thesis - Universidade de São Paulo (USP). Instituto de Matemática e Estatística (IME/SBI) São Paulo.