  Catalyzed symmetry breaking arises from a parametric enhancement of critical
fluctuations independently of the coupling strength. Symmetry-breaking
fermionic long-range fluctuations exhibit such an enhancement on negatively
curved spaces, as is known from mean-field studies. We study gravitational
catalysis from the viewpoint of the functional renormalization group using the
3d Gross-Neveu model as a specific example. We observe gravitational catalysis
towards a phase of broken discrete chiral symmetry both on a maximally
symmetric (AdS) and on a purely spatially curved manifold for constant negative
curvature (Lobachevsky plane). The resulting picture for gravitational
catalysis obtained from the renormalization flow is closely related to that of
magnetic catalysis. As an application, we estimate the curvature required for
subcritical systems of finite length to acquire a gravitionally catalyzed gap.
