Global Convergence of the Curvature Flow on Compact Hyperbolic Surfaces

This work studies a curvature flow in the conformal class of compact hyperbolic surfaces. The main theorem proves that the flow is globally defined for every admissible initial datum and converges to the unique constant-curvature metric in the class. In addition to global existence and convergence in H², the paper establishes eventual exponential decay. The argument combines global a priori bounds, elliptic bootstrap, classification of critical points via holomorphic vector field obstruction, and a spectral gradient inequality based on the Jacobi operator.