Minimal submanifolds defined by first-order systems of PDE Academic Article uri icon

abstract

  • We study first-order PDE systems implying the second-order system for minimal submanifolds of a Euclidean n-space Rn.We approach the problem geometrically by studying subsets Σ of the Grassmannian which we call m-subsets, where we define Σ to be an m-subset if all submanifolds of Rn, whose Gauss map's image is contained in Σ, are automatically minimal, m-subsets generalize the faces of calibrations studied by Harvey and Lawson. We also study linear first-order systems implying Laplace's equation, the infinitesimal version of the m-subset problem. Results include new examples of classes of minimal submanifolds admitting 'Weierstrass type' presentations in terms of holomorphic data; dimension restriction and rigidity theorems for m-subsets that extend to faces of calibrations; and showing certain codimension-two minimal submanifolds of Rn are stable using a nonconstant coefficient calibration argument. © 1992, International Press of Boston, Inc. All Rights Reserved.

author list (cited authors)

  • Landsberg, J. M.

citation count

  • 1

publication date

  • January 1992