Paper · Nov 7, 2018

Regularized mean curvature flow for invariant hypersurfaces in a Hilbert space and its application to gauge theory

Naoyuki Koike

math.DGWider flowsarXiv:1811.03441v11
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Abstract

In this paper, we investigate a regularized mean curvature flow starting from an invariant hypersurface in a Hilbert space equipped with an isometric and almost free action of a Hilbert Lie group whose orbits are minimal regularizable submanifolds. We prove that, if the initial invariant hypersurface satisfies a certain kind of horizontally convexity condition and some additional conditions, then it collapses to an orbit of the Hilbert Lie group action along the regularized mean curvature flow. In the final section, we state a vision for applying the study of the regularized mean curvature flow to the gauge theory.

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Submitted
Nov 7, 2018
Last revised
Nov 23, 2022 (v11)
Subjects
math.DG

Comments: 71 pages. arXiv admin note: text overlap with arXiv:1210.2539, arXiv:1607.02899

Mean curvature flowHomogeneous spaces

Its neighbourhood

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Connected papers This paper in the centre and the 8 papers closest to it around it, in date order clockwise from the top. The closer a paper sits to the centre, the stronger the connection. This paper Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type (Naoyuki Koike, 2016)7Koike 2016 Collapse of the mean curvature flow for certain kind of invariant hypersurfaces in a Hilbert space (Naoyuki Koike, 2016)1Koike 2016 Mean curvature flow for pinched submanifolds in rank one symmetric spaces (Naoyuki Koike, Yoshiyuki Mizumura and Nana Uenoyama, 2017)6Koike 2017 The preservability of the curvature-adaptedness along the mean curvature flow (Naoyuki Koike, 2020)5Koike 2020 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)2Fujii 2023 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)8Fujihara 2024 Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces (Naoyuki Koike and Sakura Nakaoka, 2025)3Koike 2025 Cohomogeneity one Hermann actions and backward mean curvature flows (Kurando Baba and Naoyuki Koike, 2026)4Baba 2026
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    Topics Homogeneous spaces · Same author Naoyuki Koike

  3. Mean curvature flow for principal orbits of Hermann actions on rank two symmetric spaces

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  4. Cohomogeneity one Hermann actions and backward mean curvature flows

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    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “flow starting”

  6. Mean curvature flow for pinched submanifolds in rank one symmetric spaces

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    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “collapse”, “flow starting”

  7. Volume-preserving mean curvature flow for tubes in rank one symmetric spaces of non-compact type

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    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “flow starting”

  8. Isotropy invariant graphical mean curvature flows in warped products

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Read around it

Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Homogeneous spaces On homogeneous spaces Ricci flow reduces to an ordinary differential equation on a finite-dimensional space of invariant metrics. Start with: Ricci flow of locally homogeneous geometries on closed manifolds (Isenberg et al., 1992)

In the guidebook

  1. 11A.13
    Homogeneous FlowsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A computable laboratory: the flow on Thurston’s geometries.

  2. 6A.10
    Curve Shortening and the First Geometric FlowsBook 6A · The Heat Equation and Its Relatives

    Shrinking circles, grim reapers, neckpinches and Huisken’s monotonicity, seen in the plane.