Second order families of special Lagrangian 3-folds
Abstract
A second order family of special Lagrangian submanifolds of complex m-space is a family characterized by the satisfaction of a set of pointwise conditions on the second fundamental form. For example, the set of ruled special Lagrangian submanifolds of complex 3-space is characterized by a single algebraic equation on the second fundamental form. While the `generic' set of such conditions turns out to be incompatible, i.e., there are no special Lagrangian submanifolds that satisfy them, there are many interesting sets of conditions for which the corresponding family is unexpectedly large. In some cases, these geometrically defined families can be described explicitly, leading to new examples of special Lagrangian submanifolds. In other cases, these conditions characterize already known families in a new way. For example, the examples of Lawlor-Harvey constructed for the solution of the angle conjecture and recently generalized by Joyce turn out to be a natural and easily described second order family.
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Scholars@Duke
Robert Bryant
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 special features that make them difficult to treat by the standard tools of analysis, and so my approach uses ideas and techniques from the theory of exterior differential systems, a collection of tools for analyzing such PDE systems that treats them in a coordinate-free way, focusing instead on their properties that are invariant under diffeomorphism or other transformations.
I’m particularly interested in geometric structures constrained by natural conditions, such as Riemannian manifolds whose curvature tensor satisfies some identity or that supports some additional geometric structure, such as a parallel differential form or other geometric structures that satisfy some partial integrability conditions and in constructing examples of such geometric structures, such as Finsler metrics with constant flag curvature.I am also the Director of the Simons Collaboration Special Holonomy in Geometry, Analysis, and Physics, and a considerable focus of my research and that of my students is directed towards problems in this area.
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