Let $g$ be an asymptotically flat metric on $M = \mathbb{R}^3 \setminus B_1$ where $B_1$ is the unit ball.
Suppose $X$ is a smooth vector field on $M$ that is decaying exponentially and satisfies $$\Delta_{g,conf} X - fX= 0$$ where $f$ is some smooth function and $\Delta_{g,conf}$ is the conformal laplacian defined as the divergence of the traceless part of ${L}_X g$.
Does there exist a function $f$ such that $X=0$ is the only possible solution to the above? You can assume $f$ is a function of points in $M$ and/or of $X$.
If we assume in addition that $X$ vanishes on $\partial M$, then a simple integration by parts argument shows that if $f\geq0$, then $X$ must be a conformal Killing field that decays and so must be $0$.
But does it hold without the assumption that $X$ vanishes on $\partial M$? If not, what if we know that $X$ on $\partial M$ lies on a fixed finite dimensional space? (such as the space of conformal killing fields on $\partial M$.)
