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FORWARD AND INVERSE SCATTERING ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS
, 905
"... Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codi ..."
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Abstract. We study an inverse problem for a noncompact Riemannian manifold whose ends have the following properties: On each end, the Riemannian metric is assumed to be a shortrange perturbation of the metric of the form (dy) 2 + h(x, dx), h(x, dx) being the metric of some compact manifold of codimension 1. Moreover one end is exactly cylindrical, i.e. the metric is equal to (dy) 2 + h(x, dx). Given two such manifolds having the same scattering matrix on that exactly cylindrical end for all energy, we show that these two manifolds are isometric. 1.
APPROXIMATELY EINSTEIN ACH METRICS, VOLUME RENORMALIZATION, AND AN INVARIANT FOR CONTACT MANIFOLDS
, 707
"... Abstract. To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J, we prove the existence and uniqueness, modulo highorder error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g ..."
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Abstract. To any smooth compact manifold M endowed with a contact structure H and partially integrable almost CR structure J, we prove the existence and uniqueness, modulo highorder error terms and diffeomorphism action, of an approximately Einstein ACH (asymptotically complex hyperbolic) metric g on M × (−1,0). We consider the asymptotic expansion, in powers of a special defining function, of the volume of M × (−1,0) with respect to g and prove that the log term coefficient is independent of J (and any choice of contact form θ), i.e., is an invariant of the contact structure H. The approximately Einstein ACH metric g is a generalisation of, and exhibits similar asymptotic boundary behaviour to, Fefferman’s approximately Einstein complete Kähler metric g+ on strictly pseudoconvex domains. The present work demonstrates that the CRinvariant log term coefficient in the asymptotic volume expansion of g+ is in fact a contact invariant. We discuss some implications this may have for CR Qcurvature. The formal power series method of finding g is obstructed at finite order. We show that part of this obstruction is given as a oneform on H ∗. This is a new result peculiar to the partially integrable setting. 1.
Geometric Scattering Theory with Applications to Conformal and CR Geometry
"... These lectures will be an exposition of recent work on the relation between conformal invariants of a compact manifold M and the scattering operator on real conformally compact manifold X, and its complex analogue, the CRinvariants on a compact CR manifold M and the scattering operator on certain c ..."
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These lectures will be an exposition of recent work on the relation between conformal invariants of a compact manifold M and the scattering operator on real conformally compact manifold X, and its complex analogue, the CRinvariants on a compact CR manifold M and the scattering operator on certain complex manifolds X. In geometric scattering theory, these compact manifolds M are the boundary at infinity of a manifold X with a metric that degenerates on the boundary. The scattering operator S(s) on X, for real values of the spectral parameter, is a unitary operator on L2(M). In the real case, Graham and Zworski [3] identified certain conformally invariant operators on M with the residues of the poles of the scattering matrix at certain values of the spectral parameter. A parallel construction is possible for CRinvariant operators on a CR manifold. In addition, the Qcurvature of M is obtained directly from the scattering matrix. A brief outline of the lectures: (1) Lecture 1: Review of Geometric Scattering Theory: General Case. This will be based on parts of R. Melrose’s book [8]. I will discuss the idea of
3 INVERSE SCATTERING WITH PARTIAL DATA ON ASYMPTOTICALLY HYPERBOLIC MANIFOLDS
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