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It has long been conjectured that starting at a generic smooth closed embedded surface in R 3 , the mean curvature flow remains smooth until it arrives at a singularity in a neighborhood of which the flow looks like concentric spheres or cylinders. That is, the only singularities of a generic flow are spherical or cylindrical. We will address this conjecture here and in a sequel. The higher dimensional case will be addressed elsewhere. The key to showing this conjecture is to show that shrinking spheres, cylinders, and planes are the only stable self-shrinkers under the mean curvature flow. We prove this here in all dimensions. An easy consequence of this is that every singularity other than spheres and cylinders can be perturbed away.

The Grothendieck construction establishes an equivalence between fibrations, a.k.a. fibred categories and indexed categories and is one of the fundamental results of category theory. Cockett and Cruttwell introduced the notion of fibrations into the context of tangent categories and proved that the fibres of a tangent fibration inherit a tangent structure from the total tangent category. The main goal of this paper is to provide a Grothendieck construction for tangent fibrations. Our first attempt will focus on providing a correspondence between tangent fibrations and indexed tangent categories, which are collections of tangent categories and tangent morphisms indexed by the objects and morphisms of a base tangent category. We will show that this construction inverts Cockett and Cruttwell’s result, but it does not provide a full equivalence between these two concepts. In order to understand how to define a genuine Grothendieck equivalence in the context of tangent categories, inspired by Street’s formal approach to monad theory we introduce a new concept: tangent objects. We show that tangent fibrations arise as tangent objects of a suitable $2$ -category and we employ this characterisation to lift the Grothendieck construction between fibrations and indexed categories to a genuine Grothendieck equivalence between tangent fibrations and tangent indexed categories.

We characterize simply connected John domains in the plane with the aid of weak tangents of the boundary. Specifically, we prove that a bounded simply connected domain \\(D\\) is a John domain if and only if, for every weak tangent \\(Y\\) of \\( D\\), every connected component of the complement of \\(Y\\) that ``originates\" from \\(D\\) is a John domain, not necessarily with uniform constants. Our main theorem improves a result of Kinneberg (arXiv:1507.04698), who obtains a necessary condition for a John domain in terms of weak tangents but not a sufficient one. We also establish several properties of weak tangents of John domains.

This paper introduces a novel gear drive characterized by high transmission efficiency. The gear incorporates cubic curves with corresponding tangents defining the normal section tooth profile. The generation principle of this new gear drive is thoroughly established. Utilizing gear meshing theory, key parameters such as the relative motion velocity, normal vector, and meshing equation are derived. Subsequently, the fundamental principle of the conjugate curve is formulated. The methodology for generating the tooth profile is also explored. An efficiency test was conducted on the gear, and the results demonstrate a superior transmission efficiency relative to that of traditional involute gears.

Once one knows that singularities occur, one naturally wonders what the singularities are like. For minimal varieties the first answer, already known to Federer-Fleming in 1959, is that they weakly resemble cones. For mean curvature flow, by the combined work of Huisken, Ilmanen, and White, singularities weakly resemble shrinkers. Unfortunately, the simple proofs leave open the possibility that a minimal variety or a mean curvature flow looked at under a microscope will resemble one blowup, but under higher magnification, it might (as far as anyone knows) resemble a completely different blowup. Whether this ever happens is one of the most fundamental questions about singularities. It is this long standing open question that we settle here for mean curvature flow at all generic singularities and for mean convex mean curvature flow at all singularities.

We show that limt→0 eitΔ f(x) = f(x) almost everywhere for all f ∈ Hs(ℝ²) provided that s > ⅓. This result is sharp up to the endpoint. The proof uses polynomial partitioning and decoupling.

We describe the families of minimal rational curves on any complete symmetric variety, and the corresponding varieties of minimal rational tangents (VMRT). In particular, we prove that these varieties are homogeneous and that for non-exceptional irreducible wonderful varieties, there is a unique family of minimal rational curves, and hence a unique VMRT. We relate these results to the restricted root system of the associated symmetric space. In particular we answer by the negative a question of Hwang: for certain Fano wonderful symmetric varieties, the VMRT has two connected components.

We characterize simply connected John domains in the plane with the aid of weak tangents of the boundary. Specifically, we prove that a bounded simply connected domain \\(D\\) is a John domain if and only if, for every weak tangent \\(Y\\) of \\( D\\), every connected component of the complement of \\(Y\\) that ``originates\" from \\(D\\) is a John domain, not necessarily with uniform constants. Our main theorem improves a result of Kinneberg (arXiv:1507.04698), who obtains a necessary condition for a John domain in terms of weak tangents but not a sufficient one. We also establish several properties of weak tangents of John domains.