Epsilon regularity for spaces with scalar curvature lower bounds

Robin Neumayer, Carnegie Mellon University
Fine Hall 314

In this talk, we consider Riemannian manifolds with lower bounds on scalar curvature and Perelman entropy and with upper bounds on the volume on geodesic balls. Examples show that in the absence of the assumption of volume control, these spaces need not be close to Euclidean space in any metric space sense. The added almost-Euclidean upper bound on volumes of balls ensures that geodesic balls up to a definite scale are Gromov-Hausdorff close, and in fact bi-H\"{o}lder and bi-$W^{1,p}$ homeomorphic, to Euclidean balls. We also discuss a compactness and limit space structure theorem under the same assumptions.