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

Yoshihiko Matsumoto

Vol. 262 (2013), No. 2, 437–455

DOI: 10.2140/pjm.2013.262.437

Abstract

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.

Keywords

conformal manifolds, conformally Einstein manifolds, invariant differential operators, GJMS construction, Q-curvature, ambient metric, Lichnerowicz Laplacian

Mathematical Subject Classification 2010

Primary: 53A30

Secondary: 53A55

Milestones

Received: 15 February 2012

Revised: 10 January 2013

Accepted: 21 January 2013

Published: 16 April 2013