Communications in Mathematical Physics

, Volume 359, Issue 1, pp 375–426 | Cite as

Soliton Resolution for Equivariant Wave Maps on a Wormhole

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Abstract

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
$$ds^2 = -dt^2 + dr^2 + (r^2 + 1) \left ( d \theta^2 + \sin^2 \theta d\varphi^2 \right ), \quad t,r \in \mathbb{R}, (\theta,\varphi) \in \mathbb{S}^2.$$
The constant time slices are each given by a Riemannian manifold 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 \({Q_{\ell,n} : \mathbb{R} \times \mathbb{S}^2 \rightarrow \mathbb{S}^3}\) 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. Previous work by the author proved this result for the corotational case \({\ell = 1}\) and established many preliminary results that are used in the current work.

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© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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