Paper · Feb 24, 2016

Mean curvature flow of area decreasing maps between Riemann surfaces

Andreas Savas-Halilaj, Knut Smoczyk

math.DGWider flowsarXiv:1602.07595
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As on arXiv

Abstract

In this article we give a complete description of the evolution of an area decreasing map induced by its mean curvature in the situation where and are complete Riemann surfaces with bounded geometry, being compact, for which their sectional curvatures , satisfy .

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Submitted
Feb 24, 2016
Subjects
math.DG
Mean curvature flowSurfaces

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 Evolution of area-decreasing maps between two-dimensional Euclidean spaces (Felix Lubbe, 2016)2Lubbe 2016 Lagrangian mean curvature flow of Whitney spheres (Andreas Savas-Halilaj and Knut Smoczyk, 2018)7Savas-Halila 2018 Curve shortening flow on Riemann surfaces with possible ambient conic singularities (Biao Ma, 2020)5Ma 2020 Curve shortening flow on Riemann surfaces with conical singularities (Nikolaos Roidos and Andreas Savas-Halilaj, 2020)4Roidos 2020 Graphical mean curvature flow with bounded bi-Ricci curvature (Renan Assimos, Andreas Savas-Halilaj and Knut Smoczyk, 2022)1Assimos 2022 Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces (Peter M. Topping and Hao Yin, 2023)8Topping 2023 Codimension two mean curvature flow of entire graphs (Andreas Savas-Halilaj and Knut Smoczyk, 2024)3Savas-Halila 2024 Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow (Ping-Hung Lee, 2026)6Lee 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Graphical mean curvature flow with bounded bi-Ricci curvature

    Renan Assimos, Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Mean curvature flow · Same authors Andreas Savas-Halilaj, Knut Smoczyk · Shared terms “area decreasing”, “decreasing maps”, “bounded geometry”

  2. Evolution of area-decreasing maps between two-dimensional Euclidean spaces

    Felix LubbeWider flows

    Topics Surfaces · Shared terms “decreasing maps”, “area decreasing”

  3. Codimension two mean curvature flow of entire graphs

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Mean curvature flow · Same authors Andreas Savas-Halilaj, Knut Smoczyk · Shared terms “area decreasing”, “decreasing maps”

  4. Curve shortening flow on Riemann surfaces with conical singularities

    Nikolaos Roidos and Andreas Savas-HalilajWider flows

    Topics Surfaces · Same author Andreas Savas-Halilaj · Shared terms “Riemann surfaces”

  5. Curve shortening flow on Riemann surfaces with possible ambient conic singularities

    Biao MaWider flows

    Topics Surfaces · Shared terms “Riemann surfaces”

  6. Deforming area-preserving maps between surfaces by mean curvature flow coupled with Ricci flow

    Ping-Hung Lee

    Topics Surfaces · Shared terms “Riemann surfaces”

  7. Lagrangian mean curvature flow of Whitney spheres

    Andreas Savas-Halilaj and Knut SmoczykWider flows

    Topics Mean curvature flow · Same authors Andreas Savas-Halilaj, Knut Smoczyk

  8. Uniqueness of Ricci flows from nonatomic Radon measures on Riemann surfaces

    Peter M. Topping and Hao Yin

    Topics Surfaces · Shared terms “Riemann surfaces”

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) Surfaces On surfaces Ricci flow is conformal and scalar. It gives a heat-flow proof of the uniformization theorem: every closed surface converges to constant… Start with: The Ricci flow on surfaces (Hamilton, 1988)

In the guidebook

  1. 5A.5
    Uniformization and the Two-Dimensional Ricci FlowBook 5A · Complex Analysis and Conformal Geometry

    Constant-curvature metrics on surfaces, and Ricci flow as a heat equation for the conformal factor.

  2. 11A.12
    Ricci Flow on SurfacesBook 11A · The Ricci Flow: Existence and Maximum Principles

    Uniformization by Ricci flow, and Hamilton’s entropy.

  3. 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.