In this talk we investigate what kind of CR manifolds have the following local uniqueness property: If two holomorphic functions have the same modulus on the manifold they are equal (up to a constant) in the whole space. Minimal generic submanifolds have this property. We will investigate the nonminimal case and we will give a sufficient geometric property which we will call almost minimality. This condition is related to the manifold being contained in a possibly singular real analytic Levi-flat hypersurface.