Paper · Sep 6, 2021

An anisotropic inverse mean curvature flow for spacelike graphic curves in Lorentz-Minkowski plane

Ya Gao, Chenyang Liu, Jing Mao

math.DGWider flowsarXiv:2109.02191
PDF Abstract page

As on arXiv

Abstract

In this paper, we consider the evolution of spacelike graphic curves defined over a piece of hyperbola , of center at origin and radius , in the dimensional Lorentz-Minkowski plane along an anisotropic inverse mean curvature flow with the vanishing Neumann boundary condition, and prove that this flow exists for all the time. Moreover, we can show that, after suitable rescaling, the evolving spacelike graphic curves converge smoothly to a piece of hyperbola of center at origin and prescribed radius, which actually corresponds to a constant function defined over the piece of , as time tends to infinity.

Comments: This work has been announced in our previous work [arXiv:2106.05973]. 18 pages. Comments are welcome. arXiv admin note: substantial text overlap with arXiv:2104.10600, arXiv:2104.08884

Mean curvature flowOther curvature flowsInverse mean curvature flowBoundaries & non-compactFree boundary & capillaryLong-time existence & convergencePhysics

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 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 Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space ℝⁿ⁺¹₁ (Ya Gao and Jing Mao, 2021)1Gao 2021 Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition (Jorge Lira and Fernanda Roing, 2022)8Lira 2022 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)6Chai 2022 Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space (Gregório Silva Neto and Vanessa Silva, 2023)3Neto 2023 Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space (Gregório Silva Neto and Vanessa Silva, 2023)4Neto 2023 Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone (Ya Gao and Jing Mao, 2024)2Gao 2024 Capillary Orlicz-Minkowski flow in the upper half-space (Guanghan Li and Chenyang Liu, 2026)5Li 2026 The σ-inverse mean curvature flow and the generalized Penrose conjecture (Conghan Dong, 2026)7Dong 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Inverse Gauss curvature flow in a time cone of Lorentz-Minkowski space

    Ya Gao and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Physics, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “spacelike graphic”, “Lorentz Minkowski”, “evolving spacelike”

  2. Asymptotic convergence for a class of fully nonlinear inverse curvature flows in a cone

    Ya Gao and Jing MaoWider flows PDF

    Topics Free boundary & capillary, Inverse mean curvature flow, Long-time existence & convergence · Same authors Ya Gao, Jing Mao · Shared terms “Neumann”, “converges smoothly”

  3. Classification of ruled surfaces as homothetic self-similar solutions of the inverse mean curvature flow in the Lorentz-Minkowski 3-space

    Gregório Silva Neto and Vanessa SilvaWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “Lorentz Minkowski”, “dimensional Lorentz”

  4. Ruled surfaces as translating solitons of the inverse mean curvature flow in the three-dimensional Lorentz-Minkowski space

    Gregório Silva Neto and Vanessa SilvaWider flows PDF

    Topics Inverse mean curvature flow, Other curvature flows · Shared terms “Lorentz Minkowski”, “dimensional Lorentz”

  5. Capillary Orlicz-Minkowski flow in the upper half-space

    Guanghan Li and Chenyang LiuWider flows PDF

    Topics Free boundary & capillary, Long-time existence & convergence, Other curvature flows · Same author Chenyang Liu · Shared terms “Minkowski”

  6. Inverse mean curvature flow with a free boundary in hyperbolic space

    Xiaoxiang ChaiWider flows PDF

    Topics Free boundary & capillary, Inverse mean curvature flow, Other curvature flows

  7. The -inverse mean curvature flow and the generalized Penrose conjecture

    Conghan DongWider flows PDF

    Topics Inverse mean curvature flow, Physics, Boundaries & non-compact

  8. Mean curvature flow of graphs in Generalized Robertson-Walker spacetimes with perpendicular Neumann boundary condition

    Jorge Lira and Fernanda RoingWider flows PDF

    Topics Free boundary & capillary, Long-time existence & convergence, Boundaries & non-compact · Shared terms “Neumann boundary”, “boundary 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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Other curvature flows Wider flows Flows that move a hypersurface or metric by some other function of curvature: Gauss curvature (worn stones), inverse mean curvature (the Penrose… Start with: Shapes of worn stones (Firey, 1974) Inverse mean curvature flow Wider flows A hypersurface expands with normal speed 1/H, the reciprocal of its mean curvature. Huisken and Ilmanen's weak version keeps the Hawking mass… Start with: Flow of nonconvex hypersurfaces into spheres (Gerhardt, 1990) 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)

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.