  This is our second paper in a series to study gravitational instantons, i.e.
complete hyperk\"aler 4-manifolds with faster than quadratic curvature decay.
We prove two main theorems:
  1.The asymptotic rate of gravitational instantons to the standard models can
be improved automatically.
  2.Any ALF-D_k gravitational instanton must be the
Cherkis-Hitchin-Ivanov-Kapustin-Lindstr\"om-Ro\v{c}ek metric.
