Bifurcations and new uniqueness criteria for the critical points of hyperbolic derivatives

The article describes bifurcation picture for the gradient zeros in the unit disk of the hyperbolic derivative of the holomorphic function imbedded in the family of its “level lines”. The dependence of the motion of zeros on the curvature of the hyperbolic derivative allows us to extend the Poincare–Hopf theorem to construct a new class of zero uniqueness criteria as the non-negativity of the curvature-like functionals. This class contains one-parameter series of Epstein inequalities, which are the reformulations of the Behnke–Peschl condition for the special Hartogs domains. A new rigidity phenomenon occurs: the inequalities mentioned above are contensive only for certain segment of parameters.