Paper · Aug 19, 2025

Contractive transport maps from to nearly spherical surfaces with positive Ricci curvature

Jordan Serres

math.AParXiv:2508.13688
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Abstract

We prove that every nearly spherical, positively curved surface is the contractive, volume-preserving image of a round sphere. The proof combines three main tools: the Ricci flow on surfaces, the Kim-Milman construction, and a multiscale Bakry-Émery criterion.

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Submitted
Aug 19, 2025
Subjects
math.AP, math.DG, math.FA
Ricci flowSurfacesLimit spaces & RCD

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 Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature (Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al., 2022)5Khan 2022 Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry (Georgios Papadopoulos and Edward Witten, 2024)6Papadopoulos 2024 Path Connectivity of Anosov Metrics on Surfaces (Guilherme Brandão Guglielmo and R. Ruggiero, 2026)7Guglielmo 2026 Weighted volume comparison and monotonicity for L^p-bound of Bakry-Émery Ricci curvature (Jintao Ye and Xiaohua Zhu, 2026)8Ye 2026 Spectral Obstructions to Contracting Transport Maps on Curved Spaces (Shrey Aryan, 2026)2Aryan 2026 Contracting Transport Maps on Riemannian Manifolds (Shrey Aryan, Bang-Xian Han and Zhuo-Nan Zhu, 2026)3Aryan 2026 Uniqueness of positively curved ancient Ricci flows on surfaces with boundary (Kyeongho Bang, Eric Chen and Wenkui Du, 2026)4Bang 2026 Variations of the harmonic entropy and linear drift on negatively curved surfaces (Karen Butt, Alena Erchenko and Tristan Humbert, 2026)1Butt 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Variations of the harmonic entropy and linear drift on negatively curved surfaces

    Karen Butt, Alena Erchenko and Tristan Humbert

    Topics Surfaces · Shared terms “curved surfaces”

  2. Spectral Obstructions to Contracting Transport Maps on Curved Spaces

    Shrey AryanWider flows

    Topics Limit spaces & RCD · Shared terms “transport map”, “positive Ricci”, “round sphere”

  3. Contracting Transport Maps on Riemannian Manifolds

    Shrey Aryan, Bang-Xian Han and Zhuo-Nan ZhuWider flows

    Topics Surfaces · Shared terms “transport map”, “Milman”, “round sphere”

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

    Kyeongho Bang, Eric Chen and Wenkui Du

    Topics Surfaces · Shared terms “positively curved”

  5. Log-Concavity and Fundamental Gaps on Surfaces of Positive Curvature

    Gabriel Khan, Xuan Hien Nguyen, Malik Tuerkoen et al.

    Topics Surfaces · Shared terms “curved surfaces”, “positively curved”

  6. Scale and Conformal Invariance in 2d Sigma Models, with an Application to N=4 Supersymmetry

    Georgios Papadopoulos and Edward Witten

    Topics Surfaces, Limit spaces & RCD

  7. Path Connectivity of Anosov Metrics on Surfaces

    Guilherme Brandão Guglielmo and R. Ruggiero

    Topics Surfaces

  8. Weighted volume comparison and monotonicity for -bound of Bakry-Émery Ricci curvature

    Jintao Ye and Xiaohua Zhu

    Topics Limit spaces & RCD · Shared terms “Bakry Émery”

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) Limit spaces & RCD What manifolds look like in the limit when only a lower Ricci bound is kept. Cheeger–Colding theory describes Ricci limit spaces, and RCD spaces… Start with: On the structure of spaces with Ricci curvature bounded below. I (Cheeger et al., 1997)

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. 6C.11
    Optimal Transport and Ricci CurvatureBook 6C · Nonlinear PDE: Variational and Geometric Methods

    Kantorovich duality, Wasserstein geometry, the heat equation as a gradient flow of entropy, displacement convexity, and Ricci curvature as convexity, from Lott–Sturm–Villani to McCann–Topping and the reduced volume.