a b s t r a c tWe prove that an isometric immersion of a simply connected Lorentzian surface in R 2,2 is equivalent to a normalised spinor field solution of a Dirac equation on the surface. Using the quaternions and the Lorentz numbers, we also obtain an explicit representation formula of the immersion in terms of the spinor field. We then apply the representation formula in R 2,2 to give a new spinor representation formula for Lorentzian surfaces in 3-dimensional Minkowski space. Finally, we apply the representation formula to the local description of the flat Lorentzian surfaces with flat normal bundle and regular Gauss map in R 2,2 , and show that these surfaces locally depend on four real functions of one real variable, or on one holomorphic function together with two real functions of one real variable, depending on the sign of a natural invariant.

Abstract. We prove that an isometric immersion of a simply connected Riemannian surface M in four-dimensional Minkowski space, with given normal bundle E and given mean curvature vector H ∈ Γ(E), is equivalent to a normalized spinor field ϕ ∈ Γ(ΣE ⊗ ΣM ) solution of a Dirac equation Dϕ = H · ϕ on the surface. Using the immersion of the Minkowski space into the complex quaternions, we also obtain a representation of the immersion in terms of the spinor field. We then use these results to describe the flat spacelike surfaces with flat normal bundle and regular Gauss map in four-dimensional Minkowski space, and also the flat surfaces in three-dimensional hyperbolic space, giving spinorial proofs of results by J.A. Gálvez, A. Martínez and F. Milán.

In this paper we give a geometrically invariant spinorial representation of surfaces in four-dimensional space forms. In the Euclidean space, we obtain a representation formula which generalizes the Weierstrass representation formula of minimal surfaces. We also obtain as particular cases the spinorial characterizations of surfaces in R 3 and in S 3 given by T. Friedrich and by B. Morel.

We study the Dirichlet problem for the fully nonlinear elliptic partial differential equation of second order expressing the prescription of the m th symmetric function of the principal curvatures of a spacelike hypersurface in the Minkowski space R n,1 . We completely solve the prescribed lorentzian scalar curvature equation (m = 2) in ambiant dimension 4, if the datas are strictly convex. (2000): 35B45, 35J65, 53C50. Mathematics Subject ClassificationThe aim of this article is to study the Dirichlet problem for the second order fully nonlinear partial differential equation expressing the prescription of the m th symmetric function of the principal curvatures of a spacelike hypersurface in the Minkowski space R n,1 .The corresponding problem in the euclidean context was extensively studied: first by J. Serrin [16] who discovered necessary and sufficient conditions for the solvability of the Dirichlet problem for the (quasilinear) prescribed mean curvature equation, then by N.S. Trudinger and J. Urbas [17] who studied the prescribed Gauss curvature equation (Monge-Ampère type), and lastly by N.M. Ivochkina [11,12] and N.S. Trudinger [18,19] who independently extended these results for the other m th curvature equations.In the lorentzian context, R. Bartnik and L. Simon [3] first solved the Dirichlet problem for the prescribed mean curvature equation, and Ph. Delanoë [8] that for the prescribed Gauss-Lorentz curvature equation. The remarkable fact is that euclidean obstructions disappear in the lorentzian context. C. Gerhardt [9] and O. Schnürer [15] considered prescribed curvature problems in lorentzian manifolds with structure conditions which exclude the m th curvature when 1 < m < n.Indeed, new difficulties arise in the lorentzian context with 1 < m < n, quite especially to obtain a maximum principle for the second derivatives of the solutions, which is a crucial step in theories of existence of classical solutions. Our main result is the solvability of the Dirichlet problem for the prescribed scalar curvature (m = 2) in the ambiant Minkowski space of dimension 4 when the domain and the

We prove existence and stability of smooth entire strictly convex spacelike hypersurfaces of prescribed Gauß curvature in Minkowski space. The proof is based on barrier constructions and local a priori estimates. Brought to you by | University of Connecticut Authenticated Download Date | 5/27/15 7:28 AM 2 Bayard and Schnü rer, Hypersurfaces of constant Gauß curvature in Minkowski space Brought to you by | University of Connecticut Authenticated Download Date | 5/27/15 7:28 AM