Paper · Jul 8, 2020

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

Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen, Jorge H. de Lira

math.DGWider flowsarXiv:2007.03928v5
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Abstract

Assuming that there exists a translating soliton with speed in a domain and with prescribed contact angle on , we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to as . We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of and Ricci curvature in .

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Submitted
Jul 8, 2020
Last revised
Mar 13, 2022 (v5)
Subjects
math.DG
Journal
Analysis and Geometry in Metric Spaces, vol. 10, no. 1, 2022, pp. 31-39
DOI
10.1515/agms-2020-0132

Comments: This replaces the previous versions. We have added Remark 1.2, Theorem 1.3 and its proof in Section 3

Mean curvature flowSelf-shrinkers & translatorsFree boundary & capillary

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 Minimizing movements for mean curvature flow of droplets with prescribed contact angle (Giovanni Bellettini and Shokhrukh Yusufovich Kholmatov, 2016)4Bellettini 2016 Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold M²×ℝ (Li Chen, Dan-Dan Hu, Jing Mao et al., 2018)1Chen 2018 Jenkins-Serrin problem for translating horizontal graphs in M ×ℝ (Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira et al., 2019)5Gama 2019 Translating solitons over Cartan-Hadamard manifolds (Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen et al., 2020)7Casteras 2020 A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder (Zhenghuan Gao, Bendong Lou and Jinju Xu, 2022)2Gao 2022 Consistency of minimizing movements with smooth mean curvature flow of droplets with prescribed contact-angle in ℝ³ (Shokhrukh Kholmatov, 2024)6Kholmatov 2024 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)8Eto 2024 Existence of weak mean curvature flow with prescribed contact angle via elliptic regularization (Kiichi Tashiro, 2024)3Tashiro 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Translating surfaces of the non-parametric mean curvature flow in Lorentz manifold

    Li Chen, Dan-Dan Hu, Jing Mao et al.Wider flows

    Topics Free boundary & capillary, Self-shrinkers & translators · Shared terms “parametric mean”, “prescribed contact”, “contact angle”

  2. A Mean Curvature Flow with Prescribed Contact Angles in a High Dimensional Cylinder

    Zhenghuan Gao, Bendong Lou and Jinju XuWider flows

    Topics Free boundary & capillary, Self-shrinkers & translators · Shared terms “prescribed contact”, “contact angle”

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

    Kiichi TashiroWider flows

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

  4. Minimizing movements for mean curvature flow of droplets with prescribed contact angle

    Giovanni Bellettini and Shokhrukh Yusufovich KholmatovWider flows

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

  5. Jenkins-Serrin problem for translating horizontal graphs in

    Eddygledson S. Gama, Esko Heinonen, Jorge H. de Lira et al.Wider flows

    Topics Self-shrinkers & translators · Same authors Esko Heinonen, Jorge H. de Lira · Shared terms “Riemannian product”, “translating solitons”

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

    Shokhrukh KholmatovWider flows

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

  7. Translating solitons over Cartan-Hadamard manifolds

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

    Topics Self-shrinkers & translators · Same authors Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen · Shared terms “translating solitons”, “graphical”

  8. 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 · Shared terms “prescribed contact”, “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 in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) 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. 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.