Soliton resolution for equivariant wave maps on a wormhole: I

Abstract

In this paper, we initiate the study of finite energy equivariant wave maps from the (1+3)-dimensional spacetime $\mathbb{R}\times (\mathbb{R}\times {\mathbb{S}}^2)\to {\mathbb{S}}^3$ 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 θ^2 + \sin^2 θd φ^2 \right ), \quad t,r \in \mathbb{R}, (θ,φ) \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 θ^2 + \sin^2 θd φ^2 \right ). The Riemannian manifold $\mathcal{M}$ contains two asymptotically Euclidean ends at $r\to \pm \infty$ 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 $Q_n:\mathcal{M}\to {\mathbb{S}}^3$ 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.

64 pages

References in corpus (3)

• Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrodinger equation in the radial case
• Visualizing Interstellar's Wormhole
• Wave maps on a wormhole

Cited by in corpus (2)

• Sine-Gordon on a wormhole
• Fluid Dynamics in the Ellis Wormhole