Vol. 7, No. 5, 2014

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Quasimodes and a lower bound on the uniform energy decay rate for Kerr–AdS spacetimes

Gustav Holzegel and Jacques Smulevici

Vol. 7 (2014), No. 5, 1057–1090

We construct quasimodes for the Klein–Gordon equation on the black hole exterior of Kerr–AdS (anti- de Sitter) spacetimes. Such quasimodes are associated with time-periodic approximate solutions of the Klein–Gordon equation and provide natural candidates to probe the decay of solutions on these backgrounds. They are constructed as the solutions of a semiclassical nonlinear eigenvalue problem arising after separation of variables, with the (inverse of the) angular momentum playing the role of the semiclassical parameter. Our construction results in exponentially small errors in the semiclassical parameter. This implies that general solutions to the Klein Gordon equation on Kerr–AdS cannot decay faster than logarithmically. The latter result completes previous work by the authors, where a logarithmic decay rate was established as an upper bound.

wave equation, black holes, decay estimates, Kerr – anti-de Sitter
Mathematical Subject Classification 2010
Primary: 58J50
Secondary: 83C57
Received: 3 June 2013
Revised: 24 January 2014
Accepted: 20 April 2014
Published: 27 September 2014
Gustav Holzegel
Department of Mathematics
Imperial College London
South Kensington Campus
London SW7 2AZ
United Kingdom
Jacques Smulevici
Laboratoire de Mathématiques
Université Paris Sud 11
bât. 425
91405 Orsay