Stable Soliton Resolution for Wave Maps on a Curved Spacetime

In this thesis we study finite energy equivariant wave maps posed on the (1+3)--dimensional spherically symmetric static 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 ,\begin{align*} ,ds^2 = -dt^2 + dr^2 + (r^2 + 1) \bigl ( d \theta^2 + \sin^2 \theta d \varphi^2 \bigr ), \quad t,r \in \mathbb{R}, ,(\theta,\varphi) \in \mathbb{S}^2. ,\end{align*},The metric is asymptotically flat with two ends at $r = \pm \infty$ which,are connected by a spherical ``throat" of area $4 \pi^2$ at $r = 0$. The above spacetime is often cited as a simple example ,of a wormhole geometry in general relativity but is not expected to exist in nature due to the negative energy density required to obtain it. ,We consider equivariant wave maps from the previously described spacetime into the 3--sphere, $\mathbb{s}^3$. Each equivariant wave map can be indexed by its equivariance class $\ell \in \mathbb N$ and topological degree $n \in \mathbb N \cup \{0\}$. For each $\ell$ and $n$, we prove that there exists ,a unique energy minimizing $\ell$--equivariant harmonic map $Q_{\ell,n} : \mathbb R \times (\mathbb R \times \mathbb{S}^2) \rightarrow \mathbb{S}^3$ of degree $n$. Based on mixed numerical and analytic evidence, Bizon and Kahl \cite{biz2} conjectured that all equivariant wave maps settle down to the harmonic map in the same equivariance and degree class by radiating off excess energy. In this thesis, we prove this conjecture rigorously and establish stable soliton resolution for this model; first for $\ell = 1$ (\emph{corotational} maps) in Chapter 2, and then for general $\ell > 1$ in Chapter 3. More precisely, we show that modulo a free ,radiation term, every $\ell$--equivariant wave map of degree $n$ converges strongly to $Q_{\ell,n}$.