Vol. 262, No. 2, 2013

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A GJMS construction for 2-tensors and the second variation of the total Q-curvature

Yoshihiko Matsumoto

Vol. 262 (2013), No. 2, 437–455

We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to that of Graham, Jenne, Mason, and Sparling. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operators is used to describe the second variation of the total Q-curvature. An explicit formula for conformally Einstein manifolds is given in terms of the Lichnerowicz Laplacian of an Einstein representative metric.

conformal manifolds, conformally Einstein manifolds, invariant differential operators, GJMS construction, Q-curvature, ambient metric, Lichnerowicz Laplacian
Mathematical Subject Classification 2010
Primary: 53A30
Secondary: 53A55
Received: 15 February 2012
Revised: 10 January 2013
Accepted: 21 January 2013
Published: 16 April 2013
Yoshihiko Matsumoto
Graduate School of Mathematical Sciences
The University of Tokyo
3-8-1 Komaba, Meguro-ku
Tokyo 153-8914