Epsilon regularity for spaces with scalar curvature lower bounds
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 almostEuclidean upper bound on volumes of balls ensures that geodesic balls up to a definite scale are GromovHausdorff close, and in fact biH\"{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.