Mathematical Research Letters

Volume 24 (2017)

Number 2

Equivariant wave maps on the hyperbolic plane with large energy

Pages: 449 – 479



Andrew Lawrie (Massachusetts Institute of Technology, Cambridge, Mass., U.S.A.)

Sung-Jin Oh (Korea Institute for Advanced Study (KIAS), Seoul, Korea)

Sohrab Shahshahani (Department of Mathematics and Statistics, University of Massachusetts, Amherst, Mass., U.S.A.)


In this paper we continue the analysis of equivariant wave maps from $2$-dimensional hyperbolic space $\mathbb{H}^2$ into surfaces of revolution $\mathcal{N}$ that was initiated in [12, 13]. When the target $\mathcal{N} = \mathbb{H}^2$ we proved in [12] the existence and asymptotic stability of a $1$-parameter family of finite energy harmonic maps indexed by how far each map wraps around the target. Here we conjecture that each of these harmonic maps is globally asymptotically stable, meaning that the evolution of any arbitrarily large finite energy perturbation of a harmonic map asymptotically resolves into the harmonic map itself plus free radiation. Since such initial data exhaust the energy space, this is the soliton resolution conjecture for this equation. The main result is a verification of this conjecture for a nonperturbative subset of the harmonic maps.

Received 17 May 2015

Accepted 24 February 2016

Published 24 July 2017