Finiteness of Fuchsian equations with unitarizable monodromy on compact Riemann surfaces

We prove that, on any compact Riemann surface, there are only finitely many meromorphic projective connections with prescribed singular points and real exponent differences whose projective monodromy is unitarizable. We also establish properness and growth estimates for holonomy, including a relative version for smooth marked families, and derive finiteness results for spherical conical metrics and ramification divisors of stable extensions.