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1. Mean curvature flow with generic initial data

   https://doi.org/10.1007/s00222-024-01258-0  

   Chodosh, Otis; Choi, Kyeongsu; Mantoulidis, Christos; Schulze, Felix (April 2024, Inventiones mathematicae)

   Abstract We show that the mean curvature flow of generic closed surfaces in$$\mathbb{R}^{3}$$ $R^3$avoids asymptotically conical and non-spherical compact singularities. We also show that the mean curvature flow of generic closed low-entropy hypersurfaces in$$\mathbb{R}^{4}$$ $R^4$is smooth until it disappears in a round point. The main technical ingredient is a long-time existence and uniqueness result for ancient mean curvature flows that lie on one side of asymptotically conical or compact shrinking solitons. 

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2. Mean curvature flow of noncompact hypersurfaces with Type-II curvature blow-up

   https://doi.org/10.1515/crelle-2017-0019  

   Isenberg, James; Wu, Haotian (September 2019, Journal für die reine und angewandte Mathematik (Crelles Journal))

   null (Ed.)

   Abstract We study the phenomenon of Type-II curvature blow-up in mean curvature flows of rotationally symmetric noncompact embedded hypersurfaces. Using analytic techniques based on formal matched asymptotics and the construction of upper and lower barrier solutions enveloping formal solutions with prescribed behavior, we show that for each initial hypersurface considered, a mean curvature flow solution exhibits the following behavior near the “vanishing” time T : (1) The highest curvature concentrates at the tip of the hypersurface (an umbilic point), and for each choice of the parameter {\gamma>\frac{1}{2}} , there is a solution with the highest curvature blowing up at the rate {(T-t)^{{-(\gamma+\frac{1}{2})}}} . (2) In a neighborhood of the tip, the solution converges to a translating soliton which is a higher-dimensional analogue of the “Grim Reaper” solution for the curve-shortening flow. (3) Away from the tip, the flow surface approaches a collapsing cylinder at a characteristic rate dependent on the parameter γ. 

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3. A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with type-II curvature blowup

   https://doi.org/10.1088/1361-6544/ac15a9  

   Garfinkle, David; Isenberg, James; Knopf, Dan; Wu, Haotian (August 2021, Nonlinearity)

   Abstract We present a numerical study of the local stability of mean curvature flow (MCF) of rotationally symmetric, complete noncompact hypersurfaces with type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs ‘numerically global’ (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviours under MCF. Firstly, there is a ‘near’ class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a type-II rate no slower than ( T − t ) −1 . Secondly, there is a ‘far’ class of initial data which lead to solutions developing a local type-I nondegenerate neckpinch under MCF. These numerical findings further suggest the existence of a ‘critical’ class of initial data which conjecturally lead to MCF of noncompact hypersurfaces forming local type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than ( T − t ) −1 . 

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4. Ancient solutions and translators of Lagrangian mean curvature flow

   https://doi.org/10.1007/s10240-023-00143-5  

   Lotay, Jason D; Schulze, Felix; Székelyhidi, Gábor (January 2024, Publications mathématiques de l'IHÉS)

   Abstract Suppose that ℳ is an almost calibrated, exact, ancient solution of Lagrangian mean curvature flow in$$\mathbf {C} ^{n}$$ $C^n$. We show that if ℳ has a blow-down given by the static union of two Lagrangian subspaces with distinct Lagrangian angles that intersect along a line, then ℳ is a translator. In particular in$$\mathbf {C} ^{2}$$ $C^2$, all almost calibrated, exact, ancient solutions of Lagrangian mean curvature flow with entropy less than 3 are special Lagrangian, a union of planes, or translators. 

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5. Polar foliations on symmetric spaces and mean curvature flow

   https://doi.org/10.1515/crelle-2022-0045  

   Liu, Xiaobo; Radeschi, Marco (August 2022, Journal für die reine und angewandte Mathematik (Crelles Journal))

   Abstract In this paper, we study polar foliations on simply connected symmetric spaces with non-negative curvature. We will prove that all such foliations are isoparametric as defined in [E. Heintze, X. Liu and C. Olmos,Isoparametric submanifolds and a Chevalley-type restriction theorem,Integrable systems, geometry, and topology,American Mathematical Society, Providence 2006, 151–190]. We will also prove a splitting theorem which, when leaves are compact, reduces the study of such foliations to polar foliations in compact simply connected symmetric spaces. Moreover, we will show that solutions to mean curvature flow of regular leaves in such foliations are always ancient solutions. This generalizes part of the results in [X. Liu and C.-L. Terng,Ancient solutions to mean curvature flow for isoparametric submanifolds,Math. Ann. 378 2020, 1–2, 289–315] for mean curvature flows of isoparametric submanifolds in spheres. 

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