Paper · Sep 23, 2025

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

Jiwoong Jang

math.APWider flowsarXiv:2509.19470
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Abstract

We study the motion of a droplet evolving by mean curvature with volume constraint and contact angle condition on a half space. We prove the existence of a global-in-time weak solution, called the flat flow. A difficulty arises when we establish the local-in-time equi-boundedness of approximate solutions and a uniform -estimate of multipliers. The difficulty is handled by conducting blowup analysis at a point in contact to a spherical cap with sharp angle.

Comments: 24 pages, 2 figures

Mean curvature flowVolume-preserving & constrained flowsSingularitiesWeak & singular flowsFree boundary & capillaryLong-time existence & convergence

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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 2021, 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 Consistency of the flat flow solution to the volume preserving mean curvature flow (Vesa Julin and Joonas Niinikoski, 2022)5Julin 2022 Flat flow solution to the mean curvature flow with volume constraint (Vesa Julin, 2023)4Julin 2023 Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in ℝ³ (Shokhrukh Kholmatov, 2024)8Kholmatov 2024 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)6Tashiro 2024 Existence of weak solutions to volume-preserving mean curvature flow with obstacles (Jiwoong Jang, 2025)1Jang 2025 Diffusion-aggregation equations and volume-preserving mean curvature flows (Jiwoong Jang and Antoine Mellet, 2025)3Jang 2025 Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain (Tokuhiro Eto and Jiwoong Jang, 2026)2Eto 2026 Area-preserving curvature flow for open curves with general contact angles on skew lines (Fuya Hiroi, 2026)7Hiroi 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Existence of weak solutions to volume-preserving mean curvature flow with obstacles

    Jiwoong JangWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows, Long-time existence & convergence · Same author Jiwoong Jang · Shared terms “multiplier”

  2. Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain

    Tokuhiro Eto and Jiwoong JangWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Same author Jiwoong Jang · Shared terms “contact angle”, “approximate solution”

  3. Diffusion-aggregation equations and volume-preserving mean curvature flows

    Jiwoong Jang and Antoine MelletWider flows PDF

    Topics Free boundary & capillary, Volume-preserving & constrained flows, Weak & singular flows · Same author Jiwoong Jang

  4. Flat flow solution to the mean curvature flow with volume constraint

    Vesa JulinWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “flat flow”, “volume constraint”

  5. Consistency of the flat flow solution to the volume preserving mean curvature flow

    Vesa Julin and Joonas NiinikoskiWider flows PDF

    Topics Volume-preserving & constrained flows, Weak & singular flows · Shared terms “flat flow”

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

    Kiichi TashiroWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”, “angle condition”

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

    Fuya HiroiWider flows PDF

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

  8. Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in

    Shokhrukh KholmatovWider flows PDF

    Topics Free boundary & capillary, Weak & singular flows · Shared terms “contact angle”

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) 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 usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

In the guidebook

  1. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.

  2. 6A.13
    Self-Similar Variables and MonotonicityBook 6A · The Heat Equation and Its Relatives

    Changing variables so that self-similar solutions stand still: convergence to the Gaussian, Giga–Kohn blow-up, Gaussian-weighted monotonicity formulas, frequency functions, and the blow-up logic of Huisken, Struwe and Perelman.

  3. 6C.8
    Monotonicity, ε-Regularity and Tangent ConesBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Monotonicity formulas for minimal surfaces and harmonic maps, small energy implies smoothness, bubble trees, tangent cones and dimension reduction, with soap films, liquid crystals and pseudolocality.