Rigidity in symmetric spaces

• Grant number: 11/07363-6
• Support Opportunities: Scholarships in Brazil - Post-Doctoral
• Effective date (Start): September 01, 2011
• Effective date (End): March 31, 2013
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Geometry and Topology
• Principal Investigator: Claudio Gorodski
• Grantee: Andrew James Clarke
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil

Abstract

I propose to continue my study of the geometry of riemannian spaces endowed with additional geometric structures. Specifically. I propose to study the submanifolds and gauge fields on some manifolds with structure group reduced from the orthogonal group. In the first case, I propose to study the minimal submanifolds of riemannian symmetric spaces. My previous results gave information on the isolation of a family of totally geodesic submanifolds. I would like to generalise these results to other families of submanifolds.Secondly, on manifolds with riemannian holonomy group contained in G_2, I propose to study the adapted minimal surfaces. I have already proven that these correspond with the pseudo-holomorphic curves of an associated twistor space. I believe that this second family of submanifolds can be more easily studied, using the techniques of symplectic geometry.Lastly, I have commenced an investigation into the gauge fields on manifolds of reduced holonomy. Firstly, I have given an ansatz for constructing solutions to the Yang-Mills equations on the G_2-manifolds constructed by Bryant and Salamon by considering the inherent symmetry of the problem. Secondly, on the Kummer surface and the generalised Kummer varieties constructed by Joyce, I have commenced a study of the instanton equations with respect to the distinguished reduced-holonomy metrics. This work remains in progress. (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)

CLARKE, ANDREW; TIPLER, CARL. Lower bounds on the modified K-energy and complex deformations. ADVANCES IN MATHEMATICS, v. 252, p. 449-470, FEB 15 2014. (11/07363-6)

CLARKE, ANDREW. Instantons on the exceptional holonomy manifolds of Bryant and Salamon. JOURNAL OF GEOMETRY AND PHYSICS, v. 82, p. 84-97, AUG 2014. (11/07363-6)