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We prove that sufficiently low-entropy closed hypersurfaces can be perturbedso that their mean curvature flow encounters only spherical and cylindricalsingularities. Our theorem applies to all closed surfaces in $\mathbb{R}^3$with entropy $\leq 2$ and to all closed hypersurfaces in $\mathbb{R}^4$ withentropy $\leq \lambda(\mathbb{S}^1 \times \mathbb{R}^2)$. When combined withrecent work of Daniels-Holgate, this strengthens Bernstein-Wang's low-entropySchoenflies-type theorem by relaxing the entropy bound to $\lambda(\mathbb{S}^1\times \mathbb{R}^2)$. Our techniques, based on a novel density drop argument,also lead to a new proof of generic regularity result for area-minimizinghypersurfaces in eight dimensions (due to Hardt-Simon and Smale).

Mean curvature flow with generic low-entropy initial data