Hermitian metrics on FF-manifolds

An FF-manifold is complex manifold with a multiplication on the holomorphic tangent bundle, which satisfies a certain integrability condition. Important examples are Frobenius manifolds and especially base spaces of universal unfoldings of isolated hypersurface singularities. This paper reviews the construction of hermitian metrics on FF-manifolds from tt∗tt∗ geometry. It clarifies the logic between several notions. It also introduces a new canonical hermitian metric. Near irreducible points it makes the manifold almost hyperbolic. This holds for the singularity case and will hopefully lead to applications there.