The entropy-degree theorem for Alexandrov spaces

Abstract

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.