Let $\gamma$ be a smooth metric on $S^2$ of positive curvature. Consider the metric $$g= dr^2 + r^2 \gamma$$ on $[1,\infty) \times S^2$.
Does there exist a nontrivial conformal Killing field vanishing at infinity? Does anyone know an explicit example?
If $\gamma$ is the standard metric $\gamma_{S^2}$ on the unit sphere, then the metric $g$ is the Euclidean metric, and we know that the conformal Killing fields are translations, dilations, and rotations, none of which vanish at infinity.
Here is what I have managed to do. After some work, I show that there exists a constant $C>0$ such that for any conformal Killing field $X$ going to infinity faster than $r^{-\tau}$, $\tau>0$,
$$\int r^{2\tau -1} (|X|^2 + r^2 |\nabla X|^2) dV \leq C \sup_{S^2}(|R_{\gamma}-2|, | D R_{\gamma}|)\int r^{2\tau -1} (|X|^2 + r^2 |\nabla X|^2) dV$$
where $dV$ is the volume form with respect to $g$ and $D$ is the connection on $(S^2,\gamma)$. In particular, if $\gamma$ is close enough to $\gamma_{S^2}$ in the $C^3$ norm, then we get that $X=0$. This will then imply that there does not exist a nontrivial conformal Killing field vanishing at infinity in the case that $\gamma$ is close enough to $\gamma_{S^2}$.
