Communications in Analysis and Geometry

Volume 30 (2022)

Number 3

An integral representation and decay for the wave equation in the extreme Kerr geometry

Pages: 657 – 688



Yunlong Zang (School of Mathematical Sciences, Yangzhou University, Yangzhou, China; University of Chinese Academy of Sciences, Beijing, China; and Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China)


We consider the Cauchy problem for the massless scalar wave equation in the extreme Kerr geometry with the smooth initial data compactly supported outside the event horizon. Firstly, we derive an integral representation of the solution as an infinite sum over the angular momentum modes, each of which is an integral of the energy variable $\omega$ on the real axis. This integral representation involves solutions of the radial and angular ordinary differential equations which arise in the separation of variables. Furthermore, based on this integral representation, we prove that the solution of every azimuthal mode decays pointwise in time in $L^\infty_{loc}$.

Received 2 April 2019

Accepted 3 September 2019

Published 14 December 2022