Soliton resolution for equivariant wave maps on a wormhole: I

摘要

In this paper, we initiate the study of finite energy equivariant wave mapsfrom 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 areeach 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 twoasymptotically Euclidean ends at $r \rightarrow \pm \infty$ that are connectedby 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 generalrelativity. In this work we will consider 1--equivariant or corotational wavemaps. 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, weshow 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 andKahl in the corotational case.