Vol. 308, No. 1, 2020

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Shift operators, residue families and degenerate Laplacians

Andreas Juhl and Bent Ørsted

Vol. 308 (2020), No. 1, 103–160
Abstract

In this paper, we introduce new aspects in conformal geometry of some very natural second-order differential operators. These operators are termed shift operators. In the flat space, they are intertwining operators which are closely related to symmetry breaking differential operators. In the curved case, they are closely connected with ideas of holography and the works of Fefferman–Graham, Gover–Waldron and one of the authors. In particular, we obtain an alternative description of the so-called residue families in conformal geometry in terms of compositions of shift operators. This relation allows easy new proofs of some of their basic properties. In addition, we derive new holographic formulas for Q-curvatures in even dimension. Since these turn out to be equivalent to earlier holographic formulas, the novelty here is their conceptually very natural proof. The overall discussion leads to a unification of constructions in representation theory and conformal geometry.

Keywords
Poincaré metrics, ambient metrics, conformal geometry, symmetry breaking operators, residue families, shift operators, GJMS operators, $Q$-curvature
Mathematical Subject Classification 2010
Primary: 35J30, 53A30, 53B20
Secondary: 35Q76, 53C25, 58J50
Milestones
Received: 20 August 2018
Accepted: 20 December 2019
Published: 3 December 2020
Authors
Andreas Juhl
Institut für Mathematik
Humbolt-Universität
Berlin
Germany
Bent Ørsted
Department of Mathematics
Aarhus University
Denmark