Conformal scattering on the Schwarzschild metric

We show that existing decay results for scalar fields on the Schwarzschild metric are sufficient to obtain a conformal scattering theory. Then we re-interpret this as an analytic scattering theory defined in terms of wave operators, with an explicit comparison dynamics associated with the principal null geodesic congruences. The case of the Kerr metric is also discussed.Comment: 36 pages, 6 figures. From the first version, recent references have been added and the discussion has been modified to take the new references into account. To appear in Annales de l'Institut Fourie

On Lars H\"ormander's remark on the characteristic Cauchy problem

We extend the results of a work by L. H\"ormander in 1990 concerning the resolution of the characteristic Cauchy problem for second order wave equations with regular first order potentials. The geometrical background of this work was a spatially compact spacetime with smooth metric. The initial data surface was spacelike or null at each point and merely Lipschitz. We lower the regularity hypotheses on the metric and potential and obtain similar results. The Cauchy problem for a spacelike initial data surface is solved for a Lipschitz metric and coefficients of the first order potential that are $L^\infty_\mathrm{loc}$, with the same finite energy solution space as in the smooth case. We also solve the fully characteristic Cauchy problem with very slightly more regular metric and potential, namely a ${\cal C}^1$ metric and a potential with continuous first order terms and locally $L^\infty$ coefficients for the terms of order 0.Comment: 21 pages Typing errors corrected in the estimates for the last theorem, results extended from those of the previous versio

Interacting partially directed self avoiding walk : scaling limits

This paper is dedicated to the investigation of a $1+1$ dimensional self-interacting and partially directed self-avoiding walk, usually referred to by the acronym IPDSAW and introduced in \cite{ZL68} by Zwanzig and Lauritzen to study the collapse transition of an homopolymer dipped in a poor solvant. In \cite{POBG93}, physicists displayed numerical results concerning the typical growth rate of some geometric features of the path as its length $L$ diverges. From this perspective the quantities of interest are the projections of the path onto the horizontal axis (also called horizontal extension) and onto the vertical axis for which it is useful to define the lower and the upper envelopes of the path. With the help of a new random walk representation, we proved in \cite{CNGP13} that the path grows horizontally like $\sqrt{L}$ in its collapsed regime and that, once rescaled by $\sqrt{L}$ vertically and horizontally, its upper and lower envelopes converge to some deterministic Wulff shapes. In the present paper, we bring the geometric investigation of the path several steps further. In the extended regime, we prove a law of large number for the horizontal extension of the polymer rescaled by its total length $L$, we provide a precise asymptotics of the partition function and we show that its lower and upper envelopes, once rescaled in time by $L$ and in space by $\sqrt{L}$, converge to the same Brownian motion. At criticality, we identify the limiting distribution of the horizontal extension rescaled by $L^{2/3}$ and we show that the excess partition function decays as $L^{2/3}$ with an explicit prefactor. In the collapsed regime, we identify the joint limiting distribution of the fluctuations of the upper and lower envelopes around their associated limiting Wulff shapes, rescaled in time by $\sqrt{L}$ and in space by $L^{1/4}$.Comment: 52 pages, 4 figure

The characteristic Cauchy problem for Dirac fields on curved backgrounds

On arbitrary spacetimes, we study the characteristic Cauchy problem for Dirac fields on a light-cone. We prove the existence and uniqueness of solutions in the future of the light-cone inside a geodesically convex neighbourhood of the vertex. This is done for data in $L^2$ and we give an explicit definition of the space of data on the light-cone producing a solution in $H^1$. The method is based on energy estimates following L. H\"ormander (J.F.A. 1990). The data for the characteristic Cauchy problem are only a half of the field, the other half is recovered from the characteristic data by integration of the constraints, consisting of the restriction of the Dirac equation to the cone. A precise analysis of the dynamics of light rays near the vertex of the cone is done in order to understand the integrability of the constraints; for this, the Geroch-Held-Penrose formalism is used.Comment: 39 pages. An error in lemma 3.1 in the first version has been corrected. Moreover, the treatment of the constraints (restriction of the equations to the null cone) has been considerably extended and is now given in full details. To appear in Journal of Hyperbolic Differential Equation

Regularity at space-like and null infinity

We extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino-Schoen/Chrusciel-Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ``Morawetz vector field'' to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino-Schoen/Chrusciel-Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows large classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino-Schoen/Chrusciel-Delay space-times