Advances in mathematics

The theory of pseudo differential operators, discussed in § 1, is well suited for investigating various problems connected with elliptic differential equations. However, this theory fails to be adequate for studying equations of hyperbolic type, and one is then forced to examine a wider class of operators, the so-called Fourier integral operators (Egorov [1975], Hormander [1968, 1971, 1983, 1985], Kumano-go [1982], Shubin [1978], Taylor [1981], Treves [1980]).

Fourier Integral Operators

Certain conditions for a Riemannian manifold to be isometric with a sphere:Dedicated to Professor Kentaro Yano on his fiftieth birthday

The theory of gravitational lensing is reviewed from a spacetime perspective, without quasi-Newtonian approximations More precisely, the review covers all aspects of gravitational lensing where light propagation is described in terms of lightlike geodesics of a metric of Lorentzian signature It includes the basic equations and the relevant techniques for calculating the position, the shape, and the brightness of images in an arbitrary general-relativistic spacetime It also includes general theorems on the classification of caustics, on criteria for multiple imaging, and on the possible number of images The general results are illustrated with examples of spacetimes where the lensing features can be explicitly calculated, including the Schwarzschild spacetime, the Kerr spacetime, the spacetime of a straight string, plane gravitational waves, and others

https://link.springer.com/content/pdf/10.12942%2Flrr-2004-9.pdf

Gravitational Lensing from a Spacetime Perspective

Annals of mathematics studies

Partial Representations and Partial Group Algebras

A new class of solutions of the Einstein field equations in spherical symmetry is found. The new solutions are mathematically described as the metrics admitting separation of variables in area-radius coordinates. Physically, they describe the gravitational collapse of a class of anisotropic elastic materials. Standard requirements of physical acceptability are satisfied, in particular, existence of an equation of state in closed form, weak energy condition, and existence of a regular Cauchy surface at which the collapse begins. The matter properties are generic in the sense that both the radial and the tangential stresses are non-vanishing, and the kinematical properties are generic as well, since shear, expansion, and acceleration are also non-vanishing. As a test-bed for cosmic censorship, the nature of the future singularity forming at the center is analyzed as an existence problem for o.d.e. at a singular point using techniques based on comparison theorems, and the spectrum of endstates – blackholes or naked singularities – is found in full generality. Consequences of these results on the Cosmic Censorship conjecture are discussed.

New Solutions of Einstein Equations in Spherical Symmetry: The Cosmic Censor to the Court

We prove a variational principle for geodesies on a Lorentzian manifold M. admitting a timelike Killing vector field. Using this principle and standard techniques of global nonlinear analysis we establish the existence of geodesies that join two fixed points of .M, under a suitable coercivity assumption on M. Whenever M is non contractible, we also get a multiplicity result for geodesies in M. joining two fixed points.

https://www.intlpress.com/site/pub/files/_fulltext/journals/cag/1999/0007/0001/CAG-1999-0007-0001-a006.pdf

An intrinsic approach to the Geodesical Connectedness of Stationary Lorentzian Manifolds

We give a functional analytical proof of the equalitybetween the Maslov index of a semi-Riemannian geodesicand the spectral flow of the path of self-adjointFredholm operators obtained from the index form. This fact, together with recent results on the bifurcation for critical points of strongly indefinite functionals imply that each nondegenerate and nonnull conjugate (or P-focal)point along a semi-Riemannian geodesic is a bifurcation point.In particular, the semi-Riemannian exponential map is notinjective in any neighborhood of a nondegenerate conjugate point,extending a classical Riemannian result originally due to Morse and Littauer.

Paolo Piccione

Papers

Partial Representations and Partial Group Algebras

New Solutions of Einstein Equations in Spherical Symmetry: The Cosmic Censor to the Court

An intrinsic approach to the Geodesical Connectedness of Stationary Lorentzian Manifolds

Spectral Flow, Maslov Index and Bifurcation of Semi-Riemannian Geodesics

An analytical theory for Riemannian cubic polynomials