Mathematics – Differential Geometry
Scientific paper
2000-07-21
Perspectives in Riemannian Geometry, CRM Proceedings and Lecture Notes, edited by Vestislav Apostolov, Andrew Dancer, Nigel Hi
Mathematics
Differential Geometry
36 pages, latex2e, some extra comments included in response to reader questions, many typos corrected, and important reference
Scientific paper
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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