Calibrated embeddings in the special Lagrangian and coassociative cases
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of compact calibrated submanifolds with nontrivial deformation spaces.
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Phillip Griffiths Professor of Mathematics
My research concerns problems in the geometric theory of partial differential equations. More specifically, I work on conservation laws for PDE, Finsler geometry, projective geometry, and Riemannian geometry, including calibrations and the theory of holonomy. Much of my work involves or develops techniques for studying systems of partial differential equations that arise in geometric problems. Because of their built-in invariance properties, these systems often have specia