Minimal Lagrangian submanifolds via the geodesic Gauss map

Abstract

For an oriented isometric immersed submanifold of the n-sphere, the spherical Gauss map is the Legendrian immersion of its unit normal bundle into the unit sphere subbundle of the tangent bundle of the sphere, and the geodesic Gauss map projects this into the manifold of oriented geodesics (the Grassmannian of oriented 2-planes in Euclidean n+1-space), giving a Lagrangian immersion of the unit normal bundle into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of the immersion, and show that when it has conformal shape form this depends only on its mean curvature. In particular we deduce that the geodesic Gauss map of every minimal surface in the n-sphere is minimal Lagrangian. We also give simple proofs that: deformations of the immersion always correspond to Hamiltonian deformations of its geodesic Gauss map; the mean curvature vector of the geodesic Gauss map is always a Hamiltonian vector field. This extends work of Palmer on the case when the immersion is a hypersurface.