  We propose two conjectures about Ricci-flat metrics:
  Conjecture 1: A Ricci-flat projectively induced metric is flat.
  Conjecture 2: A Ricci-flat metric on an $n$-dimensional complex manifold such
that the $a_{n+1}$ coefficient of the TYZ expansion vanishes is flat.
  We verify Conjecture 1 (see Theorem 1.1) under the assumptions that the
metric is radial and stable-projectively induced and Conjecture 2 (see Theorem
1.2) for complex surfaces whose metric is either radial or complete and ALE. We
end the paper by showing, by means of the Simanca metric, that the assumption
of Ricci-flatness in Conjecture 1 and in Theorem 1.2 cannot be weakened to
scalar-flatness (see Theorem 1.3).
