Paper · Jun 28, 2021

Ricci flow starting from an embedded closed convex surface in

Jiuzhou Huang, Jiawei Liu

math.DGarXiv:2106.14779
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Abstract

In this paper, we establish the existence and uniqueness of Ricci flow that admits an embedded closed convex surface in as metric initial condition. The main point is a family of smooth Ricci flows starting from smooth convex surfaces whose metrics converge uniformly to the metric of the initial surface in intrinsic sense.

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Submitted
Jun 28, 2021
Subjects
math.DG, math.AP, math.MG

Comments: All comments are welcome

Ricci flowStability & uniqueness

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 The conical Kähler-Ricci flow with weak initial data on Fano manifold (Jiawei Liu and Xi Zhang, 2016)1Liu 2016 Cusp Kähler-Ricci flow on compact Kähler manifold (Jiawei Liu and Xi Zhang, 2017)3Liu 2017 Stability of the conical Kähler-Ricci flows on Fano manifolds (Jiawei Liu and Xi Zhang, 2019)4Liu 2019 Closed mean curvature flows with asymptotically conical singularities (Tang-Kai Lee and Xinrui Zhao, 2024)7Lee 2024 Integral curvature estimates for solutions to Ricci flow with L^p bounded scalar curvature (Jiawei Liu and Miles Simon, 2024)5Liu 2024 Stability of the generalized Lagrangian mean curvature flow in cotangent bundle (Xishen Jin and Jiawei Liu, 2024)6Jin 2024 Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle (Jiawei Liu and Xi Zhang, 2024)8Liu 2024 Regularizing property of the twisted conical Kähler-Ricci flow (Jiawei Liu, Shiyu Zhang and Xi Zhang, 2024)2Liu 2024
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. The conical Kähler-Ricci flow with weak initial data on Fano manifold

    Jiawei Liu and Xi Zhang

    Topics Stability & uniqueness · Same author Jiawei Liu

  2. Regularizing property of the twisted conical Kähler-Ricci flow

    Jiawei Liu, Shiyu Zhang and Xi Zhang

    Topics Stability & uniqueness · Same author Jiawei Liu

  3. Cusp Kähler-Ricci flow on compact Kähler manifold

    Jiawei Liu and Xi Zhang

    Topics Stability & uniqueness · Same author Jiawei Liu

  4. Stability of the conical Kähler-Ricci flows on Fano manifolds

    Jiawei Liu and Xi Zhang

    Topics Stability & uniqueness · Same author Jiawei Liu

  5. Integral curvature estimates for solutions to Ricci flow with bounded scalar curvature

    Jiawei Liu and Miles Simon

    Topics Ricci flow · Same author Jiawei Liu

  6. Stability of the generalized Lagrangian mean curvature flow in cotangent bundle

    Xishen Jin and Jiawei LiuWider flows

    Topics Stability & uniqueness · Same author Jiawei Liu · Shared terms “initial conditions”

  7. Closed mean curvature flows with asymptotically conical singularities

    Tang-Kai Lee and Xinrui Zhao

    Topics Ricci flow · Shared terms “embedded closed”, “flow starting”

  8. Limit behavior of the twisted conical Kähler-Ricci flow with change in the cone angle

    Jiawei Liu and Xi Zhang

    Topics Ricci flow · Same author Jiawei Liu

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.

Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Stability & uniqueness When does a flow started near a special solution stay near it, and when is a flow determined by its data? Start with: Uniqueness of the Ricci Flow on Complete Noncompact Manifolds (Chen et al., 2005)

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. 10A.12
    Curve Shortening and Minimal DiscsBook 10A · Three-Manifolds and Geometrization

    How loops shrink under curvature, how the discs they bound lose area, and Perelman's extinction estimate.

  3. 11A.1
    The Equation and Its First SolutionsBook 11A · The Ricci Flow: Existence and Maximum Principles

    A heat equation for the metric: spheres, cylinders and scaling.