Mean curvature flow with generic low-entropy initial data

We prove that sufficiently low-entropy closed hypersurfaces can be perturbed so that their mean curvature flow encounters only spherical and cylindrical singularities. Our theorem applies to all closed surfaces in R3 with entropy at most 2 and to all closed hypersurfaces in R4 with entropy at most λ(S1×R2). When combined with recent work of Daniels and Holgate, this strengthens Bernstein and Wang's low-entropy Schoenflies-type theorem by relaxing the entropy bound to λ(S1×R2). Our techniques, based on a novel density drop argument, also lead to a new proof of generic regularity result for area-minimizing hypersurfaces in eight dimensions (due to Hardt, Simon, and Smale).