In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X, g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X ...

A maximal surface S with isolated singularities in a complete flat Lorentzian 3-manifold N is said to be entire if it lifts to a (periodic) entire multigraph S̃ in L. In addition, S is called of finite type if it has finite topology, finitely many singular points and S̃ is finitely sheeted. Complete and proper maximal immersions with isolated singularities in N are entire, and entire embedded max...

The classifications of holonomy groups in Lorentzian and in Euclidean signature are quite different. A group of interest in Lorentzian signature in n dimensions is the maximal proper subgroup of the Lorentz group, Sim(n− 2). Ricci-flat metrics with Sim(2) holonomy were constructed by Kerr and Goldberg, and a single four-dimensional example with a non-zero cosmological constant was exhibited by ...

The computation of the index of the Hessian of the action functional in semi-Riemannian geometry at geodesics with two variable endpoints is reduced to the case of a fixed final endpoint. Using this observation, we give an elementary proof of the Morse Index Theorem for Riemannian geodesics with two variable endpoints, in the spirit of the original Morse’s proof. This approach reduces substanti...

Hyperbolic conservation laws posed on manifolds arise in many applications to geophysical flows andgeneral relativity. Recentworkby the author and his collaborators attempts to set the foundations for a study of weak solutions defined on Riemannian or Lorentzian manifolds and includes an investigation of the existence and qualitative behavior of solutions. The metric on the manifold may either ...

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds (X, g) which are de Sitterlike at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X ...

We revisit certain path-lifting and path-continuation properties of abstract maps as described in the work F. Browder R. Rheindboldt 1950s 1960s, apply their elegant theory to exponential maps. obtain thereby a number novel results existence multiplicity geodesics joining any two points connected affine manifold, well causal connecting causally related on Lorentzian manifold. These include gene...

A rank-n tensor on a Lorentzian manifold whose contraction with n arbitrary causal future directed vectors is non-negative is said to have the dominant property. These tensors, up to sign, are called causal tensors, and we determine their general mathematical properties in arbitrary dimension N . Then, we prove that rank-2 tensors which map the null cone on itself are causal tensors. Previously...

We consider the semilinear wave equation □gu+au4=0, a≠0, on a Lorentzian manifold (M, g) with timelike boundary. show that from knowledge of Dirichlet-to-Neumann map one can recover metric g and coefficient up to natural obstructions. Our proof rests analysis interaction distorted plane waves together scattering control argument, as well Gaussian beam solutions.