Paper · Oct 31, 2023

Singular Ricci Flows on surfaces with boundary and positive scalar curvature

Jean C. Cortissoz, Juan J. Villamarín

math.DGarXiv:2310.20555
View PDF View HTML arXiv page

As on arXiv

Abstract

We study the subsequential convergence of singular solutions to the Ricci flow with prescribed constant in space geodesic curvature on compact surfaces with boundary. Furthermore, we show that in the particular case of rotational symmetry, this convergence does not depend on the sign of the geodesic curvature of the boundary.

View PDF View HTML arXiv page
Submitted
Oct 31, 2023
Subjects
math.DG, math.AP
Ricci flowSurfacesScalar curvatureWeak & singular flowsRotational symmetry & warped products

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 Singular Ricci flows I (Bruce Kleiner and John Lott, 2014)3Kleiner 2014 On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization (Abbas M Sherif, Peter K S Dunsby and Rituparno Goswami, 2021)6Sherif 2021 Rotational symmetry of solutions of mean curvature flow coming out of a double cone II (Letian Chen, 2022)8Chen 2022 A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric (Gang Li, 2025)1Li 2025 3-Manifolds with positive scalar curvature and bounded geometry (Otis Chodosh, Yi Lai and Kai Xu, 2025)5Chodosh 2025 On weak formulations of (super) Ricci flows (Sajjad Lakzian, 2026)4Lakzian 2026 Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. II: The two-dimensional space case (Roman Cherniha and John R. King, 2026)7Cherniha 2026 Uniqueness of positively curved ancient Ricci flows on surfaces with boundary (Kyeongho Bang, Eric Chen and Wenkui Du, 2026)2Bang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric

    Gang Li

    Topics Surfaces · Shared terms “geodesic curvature”, “compact surfaces”

  2. Uniqueness of positively curved ancient Ricci flows on surfaces with boundary

    Kyeongho Bang, Eric Chen and Wenkui Du

    Topics Surfaces, Rotational symmetry & warped products · Shared terms “geodesic curvature”

  3. Singular Ricci flows I

    Bruce Kleiner and John LottThe canon

    Topics Weak & singular flows

  4. On weak formulations of (super) Ricci flows

    Sajjad Lakzian

    Topics Weak & singular flows, Scalar curvature · Shared terms “positive scalar”

  5. 3-Manifolds with positive scalar curvature and bounded geometry

    Otis Chodosh, Yi Lai and Kai XuWider flows

    Topics Weak & singular flows, Scalar curvature · Shared terms “positive scalar”

  6. On some locally symmetric embedded spaces with non-negative scalar curvature and their characterization

    Abbas M Sherif, Peter K S Dunsby and Rituparno Goswami

    Topics Rotational symmetry & warped products, Scalar curvature

  7. Nonlinear systems of PDEs admitting infinite-dimensional Lie algebras and their connection with Ricci flows. II: The two-dimensional space case

    Roman Cherniha and John R. King

    Topics Surfaces, Rotational symmetry & warped products

  8. Rotational symmetry of solutions of mean curvature flow coming out of a double cone II

    Letian ChenWider flows

    Topics Rotational symmetry & warped products, Weak & singular 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.

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) 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) 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) Weak & singular flows Running a flow when the data isn't smooth or singularities form: Ricci flows from rough initial data, through singularities and on metric measure… Start with: Deformation of C⁰ Riemannian metrics in the direction of their Ricci curvature (Simon, 2002)

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. 12C.9
    After PerelmanBook 12C · Extinction and the Proof

    Singular flows, the Smale conjecture, and Bamler’s theory.