Escape Metrics and its Applications

Abstract: Geodesics escape is widely used to study the scattering of hyperbolic equations. However, there are few progresses except in a simply connected complete Riemannian manifold with nonpositive curvature. We propose a kind of complete Riemannian metrics in $\mathbb{R}^n$, which is called as escape metrics. We expose the relationship between escape metrics and geodesics escape in $\mathbb{R}^n$. Under the escape metric $g$, we prove that each geodesic of $(\mathbb{R}^n,g)$ escapes, that is, $\lim_{t\rightarrow +\infty} |\gamma (t)|=+\infty$ for any $x\in \mathbb{R}^n$ and any unit-speed geodesic $\gamma (t)$ starting at $x$. We also obtain the geodesics escape velocity and give the counterexample that if escape metrics are not satisfied, then there exists an unit-speed geodesic $\gamma (t)$ such that $\overline{\lim}_{t\rightarrow +\infty} |\gamma (t)|<+\infty$. In addition, we establish Morawetz multipliers in Riemannian geometry to derive dispersive estimates for the wave equation on an exterior domain of $\mathbb{R}^n$ with an escape metric. More concretely, for radial solutions, the uniform decay rate of the local energy is independent of the parity of the dimension $n$. For general solutions, we prove the space-time estimation of the energy and uniform decay rate $t^{-1}$ of the local energy. It is worth pointing out that different from the assumption of an Euclidean metric at infinity in the existing studies, escape metrics are more general Riemannian metrics.