The Blowdown of Ancient Noncollapsed Mean Curvature Flows

Abstract

In this paper, we consider ancient noncollapsed mean curvature flows

M_t=\partial K_t\subset {\mathbb{R}}^{n+1}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$M_t=\partial K_t\subset \mathbb {R}^{n+1}$$\end{document}

that do not split off a line. It follows from general theory that the blowdown of any time slice,

{lim}_{\lambda \to 0}\lambda K_{t_0}

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\lim _{\lambda \rightarrow 0} \lambda K_{t_0}$$\end{document}

, is at most

n-1

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n-1$$\end{document}

dimensional. Here, we show that the blowdown is in fact at most

n-2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$n-2$$\end{document}

dimensional. Our proof is based on fine cylindrical analysis, which generalizes the fine neck analysis that played a key role in many recent papers. Moreover, we show that in the uniformly k-convex case, the blowdown is at most

k-2

\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$k-2$$\end{document}

dimensional. This generalizes the recent results from Choi et al. (Geom. Topol. 28(7):3095–3134, 2024). The results and methods developed in this paper and its sequel by Du and Zhu (Adv. Math. 479:110422, 2025) have several applications in the study of mean curvature flow through singularities in higher dimensions.

Keywords

Mean curvature flow / Singularities / Ancient solutions / 53E10

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Funding

NSERC(RGPIN-2016-04331)

Alfred P. Sloan Foundation

RIGHTS & PERMISSIONS

Peking University