Soliton resolution for equivariant wave maps on a wormhole: II

TL;DR

This paper proves the soliton resolution conjecture for equivariant wave maps on a wormhole spacetime, showing that solutions decompose into a harmonic map plus radiation, thus advancing understanding of wave map dynamics in complex geometries.

Contribution

It establishes the soliton resolution for equivariant wave maps on a wormhole, extending previous results to a broader class of equivariance and confirming a conjecture by Bizon and Kahl.

Findings

Solutions decompose into harmonic map plus radiation

Convergence to stable harmonic maps is proven

Fully resolves the soliton resolution conjecture for this model

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.