Closed twisted products and SO(p) × SO(q)-invariant special Lagrangian cones

Journal Article (Journal Article)

We study a construction we call the twisted product; in this construction higher dimensional special Lagrangian (SL) and Hamiltonian stationary cones in Cp+q (equivalently special Legendrian or contact stationary submanifolds in S2(p+q)?1) are constructed by combining such objects in Cp and Cq using a suitable Legendrian curve in S3. We study the geometry of these "twisting" curves and in particular the closing conditions for them. In combination with Carberry-McIntosh's continuous families of special Legendrian 2-tori [3] and the authors' higher genus special Legendrians [13], this yields a constellation of new SL and Hamiltonian stationary cones in Cn that are topological products. In particular, for all n sufficiently large we exhibit infinitely many topological types of SL and Hamiltonian stationary cone in Cn, which can occur in continuous families of arbitrarily high dimension. A special case of the twisted product construction yields all SO(p) × SO(q)-invariant SL cones in Cp+q. These SL cones are higher-dimensional analogues of the SO(2)-invariant SL cones constructed previously by Haskins [8, 10] and used in our gluing constructions of higher genus SL cones in C3 [13]. SO(p) × SO(q)-invariant SL cones play a fundamental role as building blocks in gluing constructions of SL cones in high dimensions [14]. We study some basic geometric features of these cones including their closing and embeddedness properties.

Full Text

Duke Authors

Cited Authors

  • Haskins, M; Kapouleas, N

Published Date

  • January 1, 2012

Published In

Volume / Issue

  • 20 / 1

Start / End Page

  • 95 - 162

Electronic International Standard Serial Number (EISSN)

  • 1944-9992

International Standard Serial Number (ISSN)

  • 1019-8385

Digital Object Identifier (DOI)

  • 10.4310/CAG.2012.v20.n1.a4

Citation Source

  • Scopus