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 ${\mathbb{R}}^3$ with entropy at most 2 and to all closed hypersurfaces in ${\mathbb{R}}^4$ with entropy at most $\mathit{\lambda }({\mathbb{S}}^1\times {\mathbb{R}}^2)$. 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 $\mathit{\lambda }({\mathbb{S}}^1\times {\mathbb{R}}^2)$. Our techniques, based on a novel density drop argument, also lead to a new proof of generic regularity result for area-minimizing hypersurfaces in eight dimensions (due to Hardt, Simon, and Smale).

We address two pressing questions in the theory of the Korteweg–de Vries (KdV) equation. First, we show the uniqueness of solutions to KdV that are merely bounded, without any further decay, regularity, periodicity, or almost periodicity assumptions. The second question, emphasized by Deift, regards whether almost periodic initial data leads to almost periodic solutions to KdV. Building on the new observation that this is false for the Airy equation, we construct an example of almost periodic initial data whose KdV evolution remains bounded, but fails to be almost periodic at a later time. Our uniqueness result ensures that the solution constructed is the unique development of this initial data.

A Riemannian cone $(C,g_C)$ is by definition a warped product $C={\mathbb{R}}^{+}\times L$ with metric $g_C=d{r}^2\oplus r^2{g}_L$, where $(L,g_L)$ is a compact Riemannian manifold without boundary. We say that C is a Calabi–Yau cone if $g_C$ is a Ricci-flat Kähler metric and if C admits a $g_C$-parallel holomorphic volume form; this is equivalent to the cross-section $(L,g_L)$ being a Sasaki–Einstein manifold. In this paper, we give a complete classification of all smooth complete Calabi–Yau manifolds asymptotic to some given Calabi–Yau cone at a polynomial rate at infinity. As a special case, this includes a proof of Kronheimer’s classification of ALE hyper-Kähler 4-manifolds without twistor theory.