NO : 21 TITLE : ‘ MINIMAL LAGRANGIAN SUBMANIFOLDS VIA THE GEODESIC GAUSS MAP ’ AUTHOR ( S ) : Dr

For an oriented isometric immersion f : M → S the spherical Gauss map is the Legendrian immersion of its unit normal bundle UM⊥ into the unit sphere subbundle of TS, and the geodesic Gauss map γ projects this into the manifold of oriented geodesics in S (the Grassmannian of oriented 2-planes in R), giving a Lagrangian immersion of UM⊥ into a Kähler-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 f , and show that when f has conformal shape form this depends only on the mean curvature of f . We deduce conditions under which γ is a minimal Lagrangian immersion. We give simple proofs that: deformations of f always correspond to Hamiltonian deformations of γ; the mean curvature vector of γ is always a Hamiltonian vector field. This extends work of Palmer on the case when M is a hypersurface.