Paper · Oct 15, 2026

Cohomogeneity one Hermann actions and backward mean curvature flows

Kurando Baba, Naoyuki Koike

math.DGWider flowsarXiv:2610.21884
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Abstract

Principal orbits of cohomogeneity one Hermann actions form a natural class of equifocal hypersurfaces in symmetric spaces of compact type. Based on Kollross's classification of cohomogeneity one actions on irreducible symmetric spaces of compact type, we obtain the time evolution of the squared norm of the shape operator and the mean curvature along the long-time solution of the backward mean curvature flows starting from their principal orbits. These results explicitly describe the authors' previous generalization of the results of Liu-Terng on backward mean curvature flows of isoparametric hypersurfaces to this class of equifocal hypersurfaces.

Comments: 31 pages

Mean curvature flowHomogeneous spacesLong-time existence & convergence

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    Topics Long-time existence & convergence · Same authors Kurando Baba, Naoyuki Koike · Shared terms “backward mean”, “equifocal hypersurfaces”, “compact type”

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    Topics Homogeneous spaces · Same author Naoyuki Koike · Shared terms “Hermann action”, “principal orbits”, “symmetric space”

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    Topics Homogeneous spaces · Same author Naoyuki Koike · Shared terms “compact type”, “symmetric space”, “cohomogeneity”

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    Topics Homogeneous spaces, Long-time existence & convergence

  5. Well-posedness for the mean curvature flow on the half-space and on bounded domains

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    Topics Homogeneous spaces, Long-time existence & convergence

  6. Isotropy invariant graphical mean curvature flows in warped products

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    Topics Mean curvature flow · Same author Naoyuki Koike · Shared terms “compact type”, “symmetric space”

  7. Quaternionic Kähler manifolds fibered by solvsolitons

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    Topics Homogeneous spaces · Shared terms “principal orbits”, “cohomogeneity”, “symmetric space”

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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. 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) Long-time existence & convergence A geometric flow either runs forever or stops because something blows up. Extension criteria say which quantity must blow up at a finite singular… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

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.