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On the global “two-sided” characteristic Cauchy problem for linear wave equations on manifolds. (English) Zbl 1404.58041

In the present paper global two-sided characteristic Cauchy problem for linear wave equations on manifolds is investigated. In Section 2 known background on second-order hyperbolic PDE on manifolds is given. In Section 3 the construction of global solutions to two-sided characteristic Cauchy problem is carried out and the regularity of the solutions is examined. In Theorem 3.8 and Corollary 3.9 two fundamental jump formulae are obtained. In Section 4 representation formulae for solutions (Proposition 4.1) and a statement on continuous dependence (Corollary 4.2) are derived. In Section 5 expansion density of a null hypersurface in a Lorentzian manifold (Theorems 5.1 and 5.3) and universality in the two-sided characteristic Cauchy problem (Theorems 5.4) are studied. In Section 6 the relation with the existing literature and with some applications to quantum field theory on curved spacetime are discussed. There are four appendices.

MSC:

58J45 Hyperbolic equations on manifolds
58J47 Propagation of singularities; initial value problems on manifolds
58Z05 Applications of global analysis to the sciences
35L15 Initial value problems for second-order hyperbolic equations
53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics
81T20 Quantum field theory on curved space or space-time backgrounds
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