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1. A Numerical Stability Analysis of Mean Curvature Flow of Noncompact Hypersurfaces with Type-II Curvature Blowup: II

   https://doi.org/10.1080/10586458.2023.2201958  

   Garfinkle, David; Isenberg, James; Knopf, Dan; Wu, Haotian (May 2023, Experimental Mathematics)
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. On the constant scalar curvature Kähler metrics (II)—Existence results.

   Chen xiuxiong and Chen Jingrui (July 2021, Journal of the American Mathematical Society)
5. Complete Logarithmic Sobolev inequality via Ricci curvature bounded below II

   https://doi.org/10.1142/S1793525321500461  

   Brannan, Michael; Gao, Li; Junge, Marius (September 2023, Journal of Topology and Analysis)

   We study the “geometric Ricci curvature lower bound”, introduced previously by Junge, Li and LaRacuente, for a variety of examples including group von Neumann algebras, free orthogonal quantum groups [Formula: see text], [Formula: see text]-deformed Gaussian algebras and quantum tori. In particular, we show that Laplace operator on [Formula: see text] admits a factorization through the Laplace–Beltrami operator on the classical orthogonal group, which establishes the first connection between these two operators. Based on a non-negative curvature condition, we obtain the completely bounded version of the modified log-Sobolev inequalities for the corresponding quantum Markov semigroups on the examples mentioned above. We also prove that the “geometric Ricci curvature lower bound” is stable under tensor products and amalgamated free products. As an application, we obtain a sharp Ricci curvature lower bound for word-length semigroups on free group factors. 

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