Paper · Jun 7, 2018

The scalar curvature flow with a flat side

Hyo Seok Jang, Ki-Ahm Lee

math.APWider flowsarXiv:1806.02895v2
View PDF View HTML arXiv page

As on arXiv

Abstract

We study the near-the-interface behavior of a compact convex scalar curvature flow with a flat side. Under suitable initial conditions on the flat side, we show that the interface propagates with a finite and non-degenerate speed until the flat side vanishes. Then we get optimal derivative estimates of the pressure-like function, optimal decay estimates of curvatures near the interface, and an Aronson-Bénilan-type curvature lower bound, from which we obtain the Hölder regularity of the ratio of the curvature to the optimal decay rate up to the free boundary. In the end, we obtain the short-time and all-time existence of the solution, smooth up to the interface.

View PDF View HTML arXiv page
Submitted
Jun 7, 2018
Last revised
Feb 28, 2019 (v2)
Subjects
math.AP

Comments: 35 pages

Other curvature flowsScalar curvatureFree 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 Mean curvature flow with free boundary - Type 2 singularities (Glen Wheeler and Valentina-Mira Wheeler, 2016)6Wheeler 2016 Translating solutions to the Gauss curvature flow with flat sides (Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm Lee, 2016)1Choi 2016 Inverse mean curvature flow with a free boundary in hyperbolic space (Xiaoxiang Chai, 2022)8Chai 2022 Gauss curvature flow with shrinking obstacle (Ki-Ahm Lee and Taehun Lee, 2023)3Lee 2023 Long time regularity of the p-Gauss curvature flow with flat side (G. Huang, X. -J. Wang and Y. Zhou, 2024)5Huang 2024 Regularity of the p-Gauss curvature flow with flat side (Genggeng Huang, Xu-Jia Wang and Yang Zhou, 2024)2Huang 2024 Smoothness up to the free boundary for the p-Laplacian evolution equation and the α-Gauss curvature flow (Albert Chau and Ben Weinkove, 2024)4Chau 2024 Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds (Liam Mazurowski and Jintian Zhu, 2025)7Mazurowski 2025
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Translating solutions to the Gauss curvature flow with flat sides

    Kyeongsu Choi, Panagiota Daskalopoulos and Ki-Ahm LeeWider flows

    Topics Free boundary & capillary, Other curvature flows · Same author Ki-Ahm Lee · Shared terms “flat side”, “degenerate”

  2. Regularity of the -Gauss curvature flow with flat side

    Genggeng Huang, Xu-Jia Wang and Yang ZhouWider flows

    Topics Other curvature flows · Shared terms “flat side”, “interface”

  3. Gauss curvature flow with shrinking obstacle

    Ki-Ahm Lee and Taehun LeeWider flows

    Topics Free boundary & capillary, Other curvature flows · Same author Ki-Ahm Lee · Shared terms “time existence”

  4. Smoothness up to the free boundary for the -Laplacian evolution equation and the -Gauss curvature flow

    Albert Chau and Ben WeinkoveWider flows

    Topics Free boundary & capillary, Other curvature flows · Shared terms “flat side”, “degenerate”, “short time”

  5. Long time regularity of the -Gauss curvature flow with flat side

    G. Huang, X. -J. Wang and Y. ZhouWider flows

    Topics Other curvature flows · Shared terms “flat side”, “interface”

  6. Mean curvature flow with free boundary - Type 2 singularities

    Glen Wheeler and Valentina-Mira WheelerWider flows

    Topics Free boundary & capillary, Other curvature flows · Shared terms “type curvature”

  7. Existence of Constant Mean Curvature Surfaces in Asymptotically Flat and Asymptotically Hyperbolic Manifolds

    Liam Mazurowski and Jintian ZhuWider flows

    Topics Free boundary & capillary, Scalar curvature, Other curvature flows

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

    Xiaoxiang ChaiWider flows

    Topics Free boundary & capillary, Other curvature flows

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.

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) Scalar curvature The trace of the Ricci tensor. Under Ricci flow its minimum never decreases and positive scalar curvature (PSC) is preserved. Start with: On the proof of the positive mass conjecture in general relativity (Schoen et al., 1979) 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. 6C.4
    The Yamabe Problem and Critical ExponentsBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Conformal changes of scalar curvature, the Yamabe functional, Aubin's threshold and Schoen's solution, concentration compactness and bubbles, and why Perelman's log-Sobolev minimisers always exist.

  2. 9B.6
    Scalar Curvature and TopologyBook 9B · Comparison, Convergence and Heat on Manifolds

    Positive scalar curvature, its obstructions, and the positive mass theorem.