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 R × (R × S 2 ) → S 3 where the metric on R × (R × S 2 ) is given byThe constant time slices are each given by the Riemannian manifold M := R × S 2 with metricThe Riemannian manifold M contains two asymptotically Euclidean ends at r → ±∞ that are connected by a spherical throat of area 4π 2 at r = 0. The spacetime R × 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 : M → 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 [3] in the corotational case.