Soliton Resolution for Equivariant Wave Maps on a Wormhole

Abstract

In this paper, 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 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. The constant time slices are each given by the Riemannian manifold $\mathcal M := \mathbb R \times \mathbb{S}^2$ with metric ds^2 = dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d \varphi^2 \right ). The Riemannian manifold $\mathcal M$ 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 $\mathbb R \times \mathcal M$ 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 $Q_{n} : \mathcal M \rightarrow \mathbb{S}^3$ 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 in the corotational case.