Paper · Apr 1, 2016

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

Naoyuki Koike

math.DGWider flowsarXiv:1604.00195v6
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Abstract

First we investigate the evolutions of the radius function and its gradient along the volume-preserving mean curvature flow starting from a tube (of nonconstant radius) over a compact closed domain of a reflective submanifold in a symmetric space under certain condition for the radius function. Next, we prove that the tubeness is preserved along the flow in the case where the ambient space is a rank one symmetric space of non-compact type (other than a (real) hyperbolic space) and the reflective submanifold is an invariant submanifold and the radius function of the initial tube is radial. Furthermore, in this case, we prove that the flow reaches to the invariant submanifold or it exists in infinite time and converges to another tube of constant mean curvature over the invariant submanifold in the -topology in infinite time.

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Submitted
Apr 1, 2016
Last revised
Apr 8, 2017 (v6)
Subjects
math.DG
Journal
Calculus of Variations and Partial Differential Equations vol. 56:66 (2017)
DOI
10.1007/s00526-017-1156-2

Comments: 62pages. arXiv admin note: substantial text overlap with arXiv:1505.01652

Mean curvature flowVolume-preserving & constrained flowsSingularitiesBoundaries & non-compact

Its neighbourhood

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The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2016, plus the canon.

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 Complete bounded λ-hypersurfaces in the weighted volume-preserving mean curvature flow (Yecheng Zhu, Yi Fang and Qing Chen, 2017)6Zhu 2017 Mean curvature flow for pinched submanifolds in rank one symmetric spaces (Naoyuki Koike, Yoshiyuki Mizumura and Nana Uenoyama, 2017)1Koike 2017 Long time behaviour of discrete volume preserving mean curvature flows (Massimiliano Morini, Marcello Ponsiglione and Emanuele Spadaro, 2020)8Morini 2020 Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space (Zheng Huang, Longzhi Lin and Zhou Zhang, 2021)3Huang 2021 Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications (Ben Lambert and Elena Mäder-Baumdicker, 2022)4Lambert 2022 Translators invariant under hyperpolar actions (Tomoki Fujii and Naoyuki Koike, 2023)7Fujii 2023 Asymptotic behavior of a diffused interface volume-preserving mean curvature flow (Matteo Bonforte, Francesco Maggi and Daniel Restrepo, 2024)5Bonforte 2024 Isotropy invariant graphical mean curvature flows in warped products (Naotoshi Fujihara and Naoyuki Koike, 2024)2Fujihara 2024
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  1. Mean curvature flow for pinched submanifolds in rank one symmetric spaces

    Naoyuki Koike, Yoshiyuki Mizumura and Nana UenoyamaWider flows

    Topics Boundaries & non-compact, Singularities · Same author Naoyuki Koike · Shared terms “symmetric spaces”, “compact type”, “infinite time”

  2. Isotropy invariant graphical mean curvature flows in warped products

    Naotoshi Fujihara and Naoyuki KoikeWider flows

    Topics Singularities · Same author Naoyuki Koike · Shared terms “compact type”, “symmetric spaces”, “infinite time”

  3. Stability of the Volume Preserving Mean Curvature Flow in Hyperbolic Space

    Zheng Huang, Longzhi Lin and Zhou ZhangWider flows

    Topics Volume-preserving & constrained flows · Shared terms “hyperbolic space”

  4. Nonlocal estimates for the Volume Preserving Mean Curvature Flow and applications

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    Topics Volume-preserving & constrained flows, Singularities

  5. Asymptotic behavior of a diffused interface volume-preserving mean curvature flow

    Matteo Bonforte, Francesco Maggi and Daniel RestrepoWider flows

    Topics Volume-preserving & constrained flows

  6. Complete bounded -hypersurfaces in the weighted volume-preserving mean curvature flow

    Yecheng Zhu, Yi Fang and Qing ChenWider flows

    Topics Volume-preserving & constrained flows

  7. Translators invariant under hyperpolar actions

    Tomoki Fujii and Naoyuki KoikeWider flows

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

  8. Long time behaviour of discrete volume preserving mean curvature flows

    Massimiliano Morini, Marcello Ponsiglione and Emanuele SpadaroWider flows

    Topics Volume-preserving & constrained flows

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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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Volume-preserving & constrained flows Wider flows Curvature flows with a constraint: subtract the average of the mean curvature and the enclosed volume stays fixed while area decreases, so spheres… Start with: On an area-preserving evolution equation for plane curves (Gage, 1986) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989)

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