No abstract

We give new examples of self-shrinking and self-expanding Lagrangian solutions to the Mean Curvature Flow (MCF). These are Lagrangian submanifolds in C n , which are foliated by (n − 1)-spheres (or more generally by minimal (n − 1)-Legendrian submanifolds of S 2n−1 ), and for which the study of the self-similar equation reduces to solving a non-linear Ordinary Differential Equation (ODE). In the self-shrinking case, we get a family of submanifolds generalising in some sense the self-shrinking curves found by Abresch and Langer.

We describe natural Kähler or para-Kähler structures of the spaces of geodesics of pseudo-Riemannian space forms and relate the local geometry of hypersurfaces of space forms to that of their normal congruences, or Gauss maps, which are Lagrangian submanifolds.The space of geodesics L ± (S n+1 p,1 ) of a pseudo-Riemannian space form S n+1 p,1 of non-vanishing curvature enjoys a Kähler or para-Kähler structure (J, G) which is in addition Einstein. Moreover, in the three-dimensional case, L ± (S n+1 p,1 ) enjoys another Kähler or para-Kähler structure (J ′ , G ′ ) which is scalar flat. The normal congruence of a hypersurface S of S n+1 p,1 is a Lagrangian submanifoldS of L ± (S n+1 p,1 ), and we relate the local geometries of S andS. In particularS is totally geodesic if and only if S has parallel second fundamental form. In the three-dimensional case, we prove thatS is minimal with respect to the Einstein metric G (resp. with respect to the scalar flat metric G ′ ) if and only if it is the normal congruence of a minimal surface S (resp. of a surface S with parallel second fundamental form); moreoverS is flat if and only if S is Weingarten.

We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space R n+2 1 , the de Sitter space dS n+2 , the anti de Sitter space AdS n+2 and the Lorentzian products S n+1 ×R and H n+1 × R of the sphere and the hyperbolic space by the real line.

We make a large use of a Weierstrass representation formula to describe a variety of Hamiltonian stationary Lagrangian surfaces. Among the examples we give are the already known tori and cones, but also simply periodic cylinders, singularities of non-conical type and branch points of any order.