What question did this study set out to answer?

This work aims to investigate a curvature flow within compact hyperbolic surfaces and its long-term behavior.

July 5, 2026Open Access

Global Convergence of the Curvature Flow on Compact Hyperbolic Surfaces

Puntos clave

This work aims to investigate a curvature flow within compact hyperbolic surfaces and its long-term behavior.
Proved global existence and convergence for every admissible initial condition.
Utilized approaches like elliptic bootstrap and spectral gradient inequalities.
Classified critical points using holomorphic vector fields.
Established that curvature flow converges to a unique constant-curvature metric.
Demonstrated global a priori bounds and eventual exponential decay.

Resumen

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.

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