Extremal entropy rigidities for Riemannian metric measure spaces

We extend the work of Ledrappier-Wang and Besson-Courtois-Gallot 's barycenter technique to apply it to RCD measure metric spaces.
We prove minimal and maximal volume entropy rigidity results and express them with a new notion of homotopic degree suitable for RCD spaces.
For example, we show that an RCD(-(N-1), N) space homotopic to a hyperbolic manifold M has total measure bounded below by M's hyperbolic
volume, and equality occurs if and only if the space is isometric to M.
Joint with Dai, Connell, Perales, Núñez Zimbrón, and Wei.