Stability of Stationary Wave Maps from a Curved Background to a Sphere

TL;DR

This paper proves the existence and stability of stationary wave maps from a curved two-dimensional sphere-like manifold to a sphere, using geometric comparison theorems to handle curvature effects.

Contribution

It extends stability results of wave maps to curved backgrounds by employing triangle comparison theorems, generalizing prior work on standard spheres.

Findings

Existence of smooth, rotationally symmetric wave maps on curved backgrounds.

Stability of these wave maps in the energy topology.

Use of geometric comparison theorems to obtain key bounds.

Abstract

We study time and space equivariant wave maps from $M\times\RR\rightarrow S^2,$ where $M$ is diffeomorphic to a two dimensional sphere and admits an action of SO(2) by isometries. We assume that metric on $M$ can be written as $dr^2+f^2(r)d\theta^2$ away from the two fixed points of the action, where the curvature is positive, and prove that stationary (time equivariant) rotationally symmetric (of any rotation number) smooth wave maps exist and are stable in the energy topology. The main new ingredient in the construction, compared with the case where $M$ is isometric to the standard sphere (considered by Shatah and Tahvildar-Zadeh \cite{ST1}), is the the use of triangle comparison theorems to obtain pointwise bounds on the fundamental solution on a curved background.