Let (M, g) be Riemannian four-manifold. Does there exist a
non-zero function f:M->R such that
(*) f^2 g is flat?
(**) f^2 g satisfies Einstein equations?
Most people know the answer to (*). Nobody (really) knows the full
answer to (**). In this talk I will provide the answer to
(***) f^2 g is Kahler for some Kahler form?