Soliton Resolution for Corotational Wave Maps on a Wormhole

Abstract

Abstract In this article, we initiate the study of finite energy 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*} {\rm d}s^2 = -{\rm d}t^2 + {\rm d}r^2 + (r^2 + 1) \left ( {\rm d} \theta^2 + \sin^2 \theta {\rm 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 the Riemannian manifold ${\mathbb R} \times {\mathbb S}^2$ with metric \begin{align*} {\rm d}s^2 = {\rm d}r^2 + (r^2 + 1) \left ( {\rm d} \theta^2 + \sin^2 \theta{\rm d} \varphi^2 \right)\!. \end{align*} This Riemannian manifold contains two asymptotically Euclidean ends at $r \rightarrow \pm \infty$ that are connected by a spherical throat of area $4 \pi^2$ at $r = 0$. The spacetime ${\mathbb R} \times ({\mathbb R} \times {\mathbb S}^2) {\rightarrow} {\mathbb S}^3$ is a simple example of a wormhole geometry in general relativity. In this work, we will consider 1-equivariant or corotational wave maps. Each corotational wave map can be indexed by its topological degree $n$. For each $n$, there exists a unique energy minimizing corotational harmonic map $Q_{n} : {\mathbb R} \times {\mathbb S}^2 \rightarrow {\mathbb S}^3$ of degree $n$. In this work, we show that modulo a free radiation term, every corotational wave map of degree $n$ converges strongly to $Q_{n}$. This resolves a conjecture made by Bizon and Kahl for the corotational case.