Paper · Feb 8, 2017

Nonparametric mean curvature type flows of graphs with contact angle conditions

Hengyu Zhou

math.DGWider flowsarXiv:1702.02449
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Abstract

In this paper we study nonparametric mean curvature type flows in which are represented as graphs over a domain in a Riemannian manifold with prescribed contact angle. The speed of is the mean curvature speed minus an admissible function . Long time existence and uniformly convergence are established if with vertical contact angle and with and . Their applications include mean curvature type equations with prescribed contact angle boundary condition and the asymptotic behavior of nonparametric mean curvature flows of graphs over a convex domain in which is a surface with nonnegative Ricci curvature.

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Submitted
Feb 8, 2017
Subjects
math.DG, math.AP

Comments: 36 pages, 2 figures, to appear in International Mathematics Research Notices

Mean curvature flowVolume-preserving & constrained flowsBoundaries & non-compactFree boundary & capillaryLong-time existence & convergenceDiscrete & graph Ricci

Its neighbourhood

Connected papers

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 Mean Curvature Type Flows of Graphs in Product Manifolds (Aijin Lin and Hengyu Zhou, 2017)1Lin 2017 Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold Mⁿ×ℝ (Ya Gao, Yi-Juan Gong and Jing Mao, 2020)3Gao 2020 Non-parametric mean curvature flow with prescribed contact angle in Riemannian products (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)7Casteras 2020 A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain (Tokuhiro Eto and Yoshikazu Giga, 2024)5Eto 2024 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)8Tashiro 2024 Existence of flat flows for volume-preserving mean curvature flow with contact angle (Jiwoong Jang, 2025)6Jang 2025 Two dimensional anisotropic mean curvature flow with contact angle condition (Can Cui and Nung Kwan Yip, 2025)2Cui 2025 Area-preserving curvature flow for open curves with general contact angles on skew lines (Fuya Hiroi, 2026)4Hiroi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Mean Curvature Type Flows of Graphs in Product Manifolds

    Aijin Lin and Hengyu ZhouWider flows

    Topics Volume-preserving & constrained flows, Long-time existence & convergence · Same author Hengyu Zhou · Shared terms “type flow”, “curvature type”, “uniformly convergence”

  2. Two dimensional anisotropic mean curvature flow with contact angle condition

    Can Cui and Nung Kwan YipWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “contact angle”, “angle condition”, “angle boundary”

  3. Translating solutions of the nonparametric mean curvature flow with nonzero Neumann boundary data in product manifold

    Ya Gao, Yi-Juan Gong and Jing MaoWider flows

    Topics Free boundary & capillary · Shared terms “nonparametric mean”, “nonnegative Ricci”, “Riemannian manifold”

  4. Area-preserving curvature flow for open curves with general contact angles on skew lines

    Fuya HiroiWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “contact angle”, “angle condition”, “boundary condition”

  5. A convergence result for a minimizing movement scheme for mean curvature flow with prescribed contact angle in a curved domain

    Tokuhiro Eto and Yoshikazu GigaWider flows

    Topics Free boundary & capillary, Boundaries & non-compact · Shared terms “contact angle”, “prescribed contact”, “angle condition”

  6. Existence of flat flows for volume-preserving mean curvature flow with contact angle

    Jiwoong JangWider flows

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Long-time existence & convergence · Shared terms “contact angle”, “angle condition”

  7. Non-parametric mean curvature flow with prescribed contact angle in Riemannian products

    Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al.Wider flows

    Topics Free boundary & capillary · Shared terms “contact angle”, “prescribed contact”

  8. Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization

    Kiichi TashiroWider flows

    Topics Free boundary & capillary · Shared terms “contact angle”, “prescribed contact”, “angle condition”

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) 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) 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) Free boundary & capillary Flows whose boundary is free to slide on a support surface, meeting it at a right angle (free boundary) or at a fixed angle (capillary). Start with: Non-parametric mean curvature evolution with boundary conditions (Huisken, 1989)

In the guidebook

  1. 6D.11
    Computing Geometric FlowsBook 6D · Diffusions, Finance and Computation

    Numerical curve shortening and mean curvature flow, level sets and phase fields, discrete Ricci flow on surfaces and in conformal mapping, rotationally symmetric neckpinches, and what computation has taught about singularities.

  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.