Soliton resolution for equivariant wave maps on a wormhole: II

Abstract

In this paper, we continue our study of equivariant \emph{wave maps on a wormhole} initiated in our companion paper. More precisely, we study finite energy $\ell$--equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$ where the metric on $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ is given by \begin{align*} ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d \varphi^2 \right ), \quad t,r \in \mathbb R, (\theta,\varphi) \in \mathbb{S}^2. \end{align*} The constant time slices are each given by a Riemannian manifold $\mathcal M$ with two asymptotically Euclidean ends at $r = \pm \infty$ that are connected by a 2--sphere at $r = 0$. The spacetime $\mathbb R \times (\mathbb R \times \mathbb{S}^2)$ has appeared in the general relativity literature as a prototype wormhole geometry (but is not expected to exist in nature). Each $\ell$--equivariant finite energy wave map can be indexed by its topological degree $n$. For each $\ell$ and $n$, there exists a unique, linearly stable energy minimizing $\ell$--equivariant harmonic map $Q_{\ell,n} : \mathcal M \rightarrow \mathbb{S}^3$ of degree $n$. In this work, we prove the soliton resolution conjecture for this model. More precisely, we show that modulo a free radiation term every $\ell$--equivariant wave map of degree $n$ converges strongly to $Q_{\ell,n}$. This fully resolves a conjecture made by Bizon and Kahl. In the companion paper, we showed this for the corotational case $\ell = 1$ and established many preliminary results that are used in the current work.