Isolated maximal surfaces in spacetime

Maximal surfaces and their implications for the ambient spacetime are studied. Our methods exploit the interplay between contact of the volume functional and energy conditions. Essentially, we find that in closed universes, maximal surfaces are unique; they maximize volume; and they yield future and past singularities.Supported in part by the National Science Foundation under grant No. PHY 70-02077A03 and by the Humboldt FoundationSupported in part by the Sonderforschungsbereich (Theoretische Mathematik) of the University of Bonn     

Global properties of Gowdy spacetimes with T3 × R topology

We study the global properties of the Gowdy metrics generated by Cauchy data on the 3-torus. We show that the boundaries of the maximal Cauchy developments of Gowdy initial data sets are always “crushing singularities” in the sense of Eardley and Smarr. This means that each solution admits a slicing in which tr K(t) (the trace of the second fundamental form induced on the surface Σ_t of the slicing) uniformly blows up as t approaches its limiting value. A theorem of Hawking shows that the maximal Cauchy development cannot extend beyond the boundary at which tr K blows up and our result shows that no singularities arise to halt the evolution until this boundary is reached. Thus each maximal Cauchy development is always as large as it can be, consistent with Hawking's theorem. We discuss the relevance of this result to the strong cosmic censorship conjecture and the question of when the crushing singularities are in fact curvature singularities.     

Perturbative prepotential and monodromies in N = 2 heterotic superstring

We discuss the prepotential describing the effective field theory of N = 2 heterotic superstring models. At the one loop-level the prepotential develops logarithmic singularities due to the appearance of charged massless states at particular surfaces in the moduli space of vector multiplets. These singularities modify the classical duality symmetry group which now becomes a representation of the fundamental group of the moduli space minus the singular surfaces. For the simplest two-moduli case, this fundamental group turns out to be a certain braid group and we determine the resulting full duality transformations of the prepotential, which are exact in perturbation theory.     

Existence of maximal surfaces in asymptotically flat spacetimes

We prove the existence of maximal surfaces in asymptotically flat spacetime satisfying an interior condition. This uses a priori estimates which can also be applied to prescribed mean curvature surfaces in cosmological spacetimes and the Dirichlet problem.     

Turbulence spectra generated by singularities

The problem of turbulence spectra generated by the singularities located on lines and planes is considered. It is shown that the frequency spectrum of fluid-surface displacements due to whitecaps (linear singularities) is scaled like a weakly turbulent Zakharov-Filonenko spectrum. The corresponding wave-vector spectrum may be highly anisotropic with a decrease in maximum, as in the Phillips spectrum. However, in the isotropic situation, the spectrum differs markedly from the Phillips form. For a highly anisotropic two-dimensional turbulence, the vorticity jumps can generate the Kraichnan power-law distribution in the region of maximal angular peak. For the isotropic distribution, the turbulence spectrum coincides with the Saffman spectrum. For the shock-generated acoustic turbulence, the spectrum has the form of the Kadomtsev-Petviashvili spectrum Eω～ ω^?2 for all spatial dimensionalities.     

Polarization singularities in 2D and 3D speckle fields

The 3D structure of randomly polarized light fields is exemplified by its polarization singularities: lines along which the polarization is purely circular (C lines) and surfaces on which the polarization is linear (L surfaces). We visualize these polarization singularities experimentally in vector laser speckle fields, and in numerical simulations of random wave superpositions. Our results confirm previous analytical predictions [M. R. Dennis, Opt. Commun. 213, 201 (2002)] regarding the statistical distribution of types of C points and relate their 2D properties to their 3D structure.     

Lax Equations, Singularities and Riemann–Hilbert Problems

The existence of singularities of the solution for a class of Lax equations is investigated using a development of the factorization method first proposed by Semenov-Tyan-Shanski? (Funct Anal Appl 17(4):259?C272, 1983) and Reymann and Semenov-Tian-Shansky (1994). It is shown that the existence of a singularity at a point t?=?t _i is directly related to the property that the kernel of a certain Toeplitz operator (whose symbol depends on t) be non-trivial. The investigation of this question involves the factorization on a Riemann surface of a scalar function closely related to the above-mentioned operator. Two examples are presented which show different aspects of the problem of computing the set of singularities of the solution to the system considered. The relation between the Riemann surfaces of the classical and Lax formulation is also considered.     

Quelques remarques sur les surfaces lagrangiennes de Givental

We give a new proof of an enumerative formula of Givental about the singularities of certain lagrangian surfaces. The method is to compare these lagrangian surfaces to complex curves which may be desingularised by blowing up. A modulo 4 formula for non-orientable surfaces is also obtained and it is shown how to construct all the Givental lagrangian embeddings by lagrangian surgery.     

Viscous shocks in Hele-Shaw flow and Stokes phenomena of the Painlevé I transcendent

In Hele-Shaw flows at vanishing surface tension, the boundary of a viscous fluid develops cusp-like singularities. In recent papers Lee et al. (2009, 2008) [8] and [9] we have showed that singularities trigger viscous shocks propagating through the viscous fluid. Here we show that the weak solution of the Hele-Shaw problem describing viscous shocks is equivalent to a semiclassical approximation of a special real solution of the Painlevé I equation. We argue that the Painlevé I equation provides an integrable deformation of the Hele-Shaw problem which describes flow passing through singularities. In this interpretation shocks appear as Stokes level-lines of the Painlevélinear problem.     

Non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane

The non-local theory solution of two collinear mode-I permeable cracks in a magnetoelectroelastic composite material plane was investigated using the generalized Almansi's theorem and the Schmidt method. The problem was formulated through Fourier transform into two pairs of dual integral equations, in which the unknown variables are the jumps in displacements across the crack surfaces. To solve the dual integral equations, the displacement jumps across the crack surfaces were directly expanded as a series of Jacobi polynomials. Numerical examples were provided to show the effects of crack length, the distance between two collinear cracks and the lattice parameter on the stress field, the electric displacement field and the magnetic flux field near the crack tips. Unlike the classical elasticity solutions, it is found that no stress, electric displacement or magnetic flux singularities are present at the crack tips in a magnetoelectroelastic composite material plane. The non-local elastic solutions yield a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion.     

Solving the relativistic inverse scattering problem with allowance for inelasticity effects on the basis of <Emphasis Type="Italic">N/D</Emphasis> equations and application of the resulting solution to an analysis of nucleon-nucleon interaction

A manifestly Poincaré-invariant approach to solving the inverse scattering problem is developed with allowance for inelasticity effects. The equations of the N/D method are used as dynamical equations in this approach. Two versions of the approach are considered. In the first version (method A), the required equations are constructed on the basis of the maximal-analyticity principle, which constitutes the basis of dynamical S-matrix theory. In formulating the second version of equations (method B), it is assumed that a partial-wave scattering amplitude may develop dynamical singularities that violate the requirement of maximal analyticity. The dynamics of interaction components that violate maximal analyticity is described within the model of a nonlocal separable potential. The method is used to analyze nucleon-nucleon interaction in the ¹S₀ and ³S₁ states. The results obtained by solving the inverse scattering problem for potential functions are compared with the predictions of the one-boson-exchange model.     

A thermodynamic discussion of the possibility of singular yield conditions in plasticity theory

A generalization is given of the author's theory for plasticity phenomena. The generalization leads to the possibility that the yield surface has singularities. From the theory a formula may be derived which is analogous to a formula proposed by Koiter for plastic flow in media with singular yield surfaces. The possibility of elastic relaxation phenomena in the preplastic range is included in the developed formalism.     

Maximal hypersurfaces and foliations of constant mean curvature in general relativity

We prove theorems on existence, uniqueness and smoothness of maximal and constant mean curvature compact spacelike hypersurfaces in globally hyperbolic spacetimes. The uniqueness theorem for maximal hypersurfaces of Brill and Flaherty, which assumed matter everywhere, is extended to spacetimes that are vacuum and non-flat or that satisfy a generic-type condition. In this connection we show that under general hypotheses, a spatially closed universe with a maximal hypersurface must be Wheeler universe; i.e. be closed in time as well. The existence of Lipschitz achronal maximal volume hypersurfaces under the hypothesis that candidate hypersurfaces are bounded away from the singularity is proved. This hypothesis is shown to be valid in two cases of interest: when the singularities are of strong curvature type, and when the singularity is a single ideal point. Some properties of these maximal volume hypersurfaces and difficulties with Avez' original arguments are discussed. The difficulties involve the possibility that the maximal volume hypersurface can be null on certain portions; we present an incomplete argument which suggests that these hypersurfaces are always smooth, but prove that an a priori bound on the second fundamental form does imply smoothness. An extension of the perturbation theorem of Choquet-Bruhat, Fischer and Marsden is given and conditions under which local foliations by constant mean curvature hypersurfaces can be extended to global ones is obtained.     

Uniqueness of maximal surfaces in Generalized Robertson–Walker spacetimes and Calabi–Bernstein type problems

Complete maximal surfaces in Generalized Robertson–Walker spacetimes obeying either the Null Convergence Condition or the Timelike Convergence Condition are studied. Uniqueness theorems that widely extend the classical Calabi–Bernstein theorem, as well as previous results on complete maximal surfaces in Robertson–Walker spacetimes, i.e. the case in which the Gauss curvature of the fiber is a constant, are given. All the entire solutions to the maximal surface differential equation in certain Generalized Robertson–Walker spacetimes are found.     

A Note on the Null Surface Formulation of GR

The purpose of this work is to study the structure and nature of the singularities of wavefronts in flat space-time. We computed the behavior at the singularities of important objects that take place in the null surface formulation of general relativity. As a secondary result we show that the Minkowski space-time with non-trivial null surfaces is a solution of the null surface approach to general relativity.     

Hyperboloid structures formed by polarization singularities in coherent vector fields with longitudinal-transverse coupling

The propagation-dependent polarization vector fields are experimentally created from an isotropic microchip laser with a longitudinal-transverse coupling and entanglement of the polarization states. The experimental three-dimensional coherent vector fields are analytically reconstructed with a coherent superposition of orthogonal circularly polarized vortex modes. Each polarized component is found to comprise two Laguerre-Gaussian modes with different topological charges. With the analytical representation, the polarization singularities, on which the electric polarization ellipse is purely circular (C lines) or purely linear (L surfaces), are explored. The C line singularities are found to form an intriguing hyperboloidal structure.     

Collisions of Particles in Locally AdS Spacetimes II Moduli of Globally Hyperbolic Spaces

We investigate globally hyperbolic 3-dimensional AdS manifolds containing “particles”, i.e., cone singularities of angles less than 2π along a time-like graph Γ. To each such space (equipped with a time-like vector field satisfying some additional properties) we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient to recover the space locally (i.e., in the neighborhood of a fixed metric). This is a partial extension of a result of Mess for non-singular globally hyperbolic AdS manifolds.