The entropy-degree theorem for Alexandrov spaces

We present the entropy-degree theorem for Lipschitz maps between Alexandrov spaces with curvature bounded below, extending the classical Besson–Courtois–Gallot entropy-rigidity results to this singular setting. The proof requires a new degree theorem for Alexandrov spaces, developed using the Ambrosio–Kirchheim theory of integral currents, showing the equivalence between analytical and topological degrees. Applications include geometric obstructions for negatively curved Einstein metrics on 4-orbifolds, volume bounds for cone-manifolds, quantitative inequalities for hyperbolic convex cores, and lower bounds on the asymptotic translation lengths of end-periodic surface homeomorphisms. We show that entropy-volume minimization under uniform lower curvature bounds obstructs to the formation of metric singularities in Gromov–Hausdorff limits, prove an Alexandrov boundary rigidity theorem, and establish volume minima for cone manifolds and cone orbifolds.