The space of complete embedded maximal surfaces with isolated singularities in the 3-dimensional Lorentz-Minkowski space

We prove that a complete embedded maximal surface in = (³, dx₁² + dx₂²-dx₃²) with a finite number of singularities is an entire maximal graph with conelike singularities over any spacelike plane, and so, it is asymptotic to a spacelike plane or a half catenoid. We show that the moduli space of entire maximal graphs over {x₃=0} in with n+12 singular points and vertical limit normal vector at infinity is a 3n+4-dimensional differentiable manifold. The convergence in means the one of conformal structures and Weierstrass data, and it is equivalent to the uniform convergence of graphs on compact subsets of {x₃=0}. Moreover, the position of the singular points in ³ and the logarithmic growth at infinity can be used as global analytical coordinates with the same underlying topology. We also introduce the moduli space of marked graphs with n+1 singular points (a mark in a graph is an ordering of its singularities), which is a (n+1)-sheeted covering of . We prove that identifying marked graphs differing by translations, rotations about a vertical axis, homotheties or symmetries about a horizontal plane, the corresponding quotient space is an analytic manifold of dimension 3n–1. This manifold can be identified with a spinorial bundle associated to the moduli space of Weierstrass data of graphs in .Mathematics Subject Classification (2000): 53C50, 58D10, 53C42First and second authors research partially supported by MEC-FEDER grant number MTM2004-00160Second and third authors research partially supported by Consejería de Educación y Ciencia de la Junta de Andalucía and the European Union.     

Higher Schläfli Formulas and Applications

The classical Schläfli formula relates the variations of the dihedral angles of a smooth family of polyhedra in a space-form to the variation of the enclosed volume. We give higher analogues of this formula: for each p, we prove a simple formula relating the variation of the volumes of the codimension p faces to the variation of the 'curvature' – the volumes of the duals of the links in the convex case – of codimension p+2 faces. It is valid also for ideal polyhedra, or for polyhedra with some ideal vertices. This extends results of Suárez-Peiró. The proof is through analoguous smooth formulas. Some applications are described.     

Mean curvature rigidity of horospheres,hyperspheres, and hyperplanes

Archiv der Mathematik - We prove that horospheres, hyperspheres, and hyperplanes in a hyperbolic space $${\mathbb {H}}^n,\,n\ge 3$$ , admit no perturbations with compact support which increase...     

The moduli space of embedded singly periodic maximal surfaces with isolated singularities in the Lorentz–Minkowski space $$\mathbb{L}^3$$

We show that, up to some natural normalizations, the moduli space of singly periodic complete embedded maximal surfaces in the Lorentz–Minkowski space , with fundamental piece having a finite number (n + 1) of singularities, is a real analytic manifold of dimension 3n + 4. The underlying topology agrees with the topology of uniform convergence of graphs on compact subsets of {x ₃ = 0}.