On the spinorial representation of spacelike surfaces into 4-dimensional Minkowski space

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 \in \Gamma(E), is equivalent to a normalized spinor field \varphi \in \Gamma(\Sigma E \otimes \Sigma M) solution of a Dirac equation D\varphi=H\cdot\varphi 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. Galvez, A. Martinez and F. Milan

Spinorial representation of submanifolds in SLn(C)/SU(n)SL_n(\mathbb{C})/SU(n)

We give a spinorial representation of a submanifold of any dimension and co-dimension in a symmetric space $G/H,$ where $G$ is a complex semi-simple Lie group and $H$ is a compact real form of $G.$ This in particular includes $SL_n(\mathbb{C})/SU(n),$ and extends the previously known spinorial representation of a surface in $\mathbb{H}^3$ if $n=2.$ We also recover the Bryant representation of a surface with constant mean curvature 1 in $\mathbb{H}^3$ and its generalization for a surface with holomorphic right Gauss map in $SL_n(\mathbb{C})/SU(n).$ As a new application, we obtain a fundamental theorem for the submanifold theory in that spaces.Comment: 34 pages, to appear in Advances in Applied Clifford Algebras (this version of the paper contains many improvements with respect to the first submission

Entire spacelike hypersurfaces of prescribed scalar curvature in Minkowski space

We prove existence and uniqueness of entire spacelike hypersurfaces in the Minkowski space with prescribed negative scalar curvature, and with given values at infinity which stay at a bounded distance of a lightcone.Comment: 21 pages; revised version; to appear in Calc. of Var. and PDE'

A spinor description of flat surfaces in R^4

We describe the flat surfaces with flat normal bundle and regular Gauss map immersed in R^4 using spinors and Lorentz numbers. We obtain a new proof of the local structure of these surfaces. We also study the flat tori in the sphere S^3 and obtain a new representation formula. We then deduce new proofs of their global structure, and of the global structure of their Gauss map image.Comment: 33 page

Entire scalar curvature flow and hypersurfaces of constant scalar curvature in Minkowski space

We prove existence in the Minkowski space of entire spacelike hypersurfaces with constant negative scalar curvature and given set of lightlike directions at infinity; we also construct the entire scalar curvature flow with prescribed set of lightlike directions at infinity, and prove that the flow converges to a spacelike hypersurface with constant scalar curvature. The proofs rely on barriers construction and a priori estimates

Entire spacelike radial graphs in the Minkowski space, asymptotic to the light-cone, with prescribed scalar curvature

Existence and uniqueness in ${\Bbb R}^{n,1}$ of entire spacelike hypersurfaces contained in the future of the origin $O$ and asymptotic to the light-cone, with scalar curvature prescribed at their generic point $M$ as a negative function of the unit vector $\overrightarrow{Om}$ pointing in the direction of $\overrightarrow{OM}$, divided by the square of the norm of $\overrightarrow{OM}$ (a dilation invariant problem). The solutions are seeked as graphs over the future unit-hyperboloid emanating from $O$ (the hyperbolic space); radial upper and lower solutions are constructed which, relying on a previous result in the Cartesian setting, imply their existence.Comment: 13 page

Spinor representation of Lorentzian surfaces in R^{2,2}

We prove that an isometric immersion of a simply connected Lorentzian surface in $\mathbb{R}^{2,2}$ 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 $\mathbb{R}^{2,2}$ 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 $\mathbb{R}^{2,2},$ 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.Comment: arXiv admin note: text overlap with arXiv:1212.3543 . Many improvements concerning the writing of the paper. To appear in Journal of Geometry and Physic

Geometric invariants and principal configurations on spacelike surfaces immersed in R^3,1

We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observations on the mean directionally curved lines and on the asymptotic lines on spacelike surfaces end the paper.Comment: 27 page

On the affine Gauss maps of submanifolds of Euclidean space

It is well known that the space of oriented lines of Euclidean space has a natural symplectic structure. Moreover, given an immersed, oriented hypersurface S the set of oriented lines that cross S orthogonally is a Lagrangian submanifold. Conversely, if \bar{S} an n-dimensional family of oriented lines is Lagrangian, there exists, locally, a 1-parameter family of immersed, oriented, parallel hypersurfaces S_t whose tangent spaces cross orthogonally the lines of \bar{S}. The purpose of this paper is to generalize these facts to higher dimension: to any point x of a submanifold S of R^m of dimension n and co-dimension k=m-n, we may associate the affine k-space normal to S at x. Conversely, given an n-dimensional family \bar{S} of affine k-spaces of R^m, we provide certain conditions granting the local existence of a family of n-dimensional submanifolds S which cross orthogonally the affine k-spaces of \bar{S}. We also define a curvature tensor for a general family of affine spaces of R^m which generalizes the curvature of a submanifold, and, in the case of a 2-dimensional family of 2-planes in R^4, show that it satisfies a generalized Gauss-Bonnet formula.Comment: 28 page

Entire spacelike hypersurfaces of constant Gauss curvature in Minkowski space

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.Comment: 31 page