Paper · Mar 3, 2025

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

Naoyuki Koike, Sakura Nakaoka

math.DGWider flowsarXiv:2503.01355
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Abstract

In this paper, we illustrate the behaviour of the mean curvature flows starting from principal orbits of any commuting Hermann action of cohomogeneity two on irreducible rank two Riemannian symmetric spaces of compact type by using Mathematicae. In more detail, we illustrate the velocity vector fields of the curves (determined by the flows) on the orbit space (which is a 2-simplex) of the Hermann action by using Mathematcae. Also, we calculate the position of the point of the orbit space corresponding to the only minimal principal orbit of the Hermann action by using Mathematicae.

Comments: 30 pages, 36 figures

Mean curvature flowHomogeneous spaces

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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 “symmetric space”, “compact type”, “cohomogeneity”

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

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

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

  6. Ricci solitons as submanifolds of complex hyperbolic spaces

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    Topics Homogeneous spaces · Shared terms “symmetric space”, “compact type”

  7. Inhomogeneous deformations of Einstein solvmanifolds

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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. 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.