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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


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)

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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, . (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, . (11/07363-6)

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