  We analyze the indicial roots of the self-dual deformation complex on a
cylinder $(\mathbb{R} \times Y^3, dt^2 + g_Y)$, where $Y^3$ is a space of
constant curvature. An application is the optimal decay rate of solutions on a
self-dual manifold with cylindrical ends having cross-section $Y^3$. We also
resolve a conjecture of Kovalev-Singer in the case where $Y^3$ is a hyperbolic
rational homology 3-sphere, and show that there are infinitely many examples
for which the conjecture is true, and infinitely many examples for which the
conjecture is false. Applications to gluing theorems are also discussed.
