Paper · Nov 2, 2026

Conjugate radius and parabolicity along the Ricci flow

Chandan Kumar Mondal

math.DGarXiv:2611.03823
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Abstract

We study the behaviour of the conjugate radius under the Ricci flow on complete noncompact manifolds. We show that if is a complete Ricci flow with nonnegative scalar curvature at , the initial metric is parabolic and not Ricci-flat, and the Einstein tensor is bounded above, then parabolicity is preserved along the flow and the conjugate radius of is finite for every . As an application, we prove that a complete noncompact gradient shrinking Ricci soliton whose volume growth is subquadratic, in the sense that , has finite conjugate radius.

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Submitted
Nov 2, 2026
Subjects
math.DG

Comments: 8 pages

Ricci flowRicci solitonsRicci-flat & Calabi–YauScalar curvatureBoundaries & non-compact

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 On a dichotomy of the curvature decay of steady Ricci soliton (Pak-Yeung Chan and Bo Zhu, 2021)8Chan 2021 Weak scalar curvature lower bounds along Ricci flow (Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang, 2021)2Jiang 2021 A comparison theorem for steady Ricci solitons (Benedito Leandro and Jeferson Poveda, 2022)5Leandro 2022 Triviality Results and Conjugate Radius Estimation of Ricci Solitons (Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu, 2023)1Shaikh 2023 L^2-instability of the Taub-Bolt metric under the Ricci flow (John Hughes, 2024)6Hughes 2024 Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature (Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al., 2024)4Li 2024 Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions (Sigurd B. Angenent and Dan Knopf, 2025)7Angenent 2025 Pseudolocality and strong uniqueness for complete Ricci flows (Shaochuang Huang and Zhuo Peng, 2026)3Huang 2026
Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other
  1. Triviality Results and Conjugate Radius Estimation of Ricci Solitons

    Absos Ali Shaikh, Prosenjit Mandal and V. Amarendra Babu

    Topics Boundaries & non-compact, Ricci solitons · Shared terms “conjugate radius”, “gradient shrinking”, “shrinking Ricci”

  2. Weak scalar curvature lower bounds along Ricci flow

    Wenshuai Jiang, Weimin Sheng and Huaiyu Zhang

    Topics Ricci-flat & Calabi–Yau, Scalar curvature · Shared terms “initial metric”, “preserved along”

  3. Pseudolocality and strong uniqueness for complete Ricci flows

    Shaochuang Huang and Zhuo Peng

    Topics Ricci-flat & Calabi–Yau · Shared terms “complete Ricci”, “initial metric”

  4. Rigidity of Five-dimensional Shrinking Gradient Ricci Solitons with Constant Scalar Curvature

    Fengjiang Li, Jianyu Ou, Yuanyuan Qu et al.

    Topics Scalar curvature, Boundaries & non-compact, Ricci solitons · Shared terms “noncompact gradient”, “complete noncompact”, “gradient shrinking”

  5. A comparison theorem for steady Ricci solitons

    Benedito Leandro and Jeferson Poveda

    Topics Ricci-flat & Calabi–Yau, Scalar curvature, Boundaries & non-compact

  6. L^2-instability of the Taub-Bolt metric under the Ricci flow

    John Hughes

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact, Ricci solitons

  7. Infinite-dimensional dynamical instabilities of noncompact stationary Ricci flow solutions

    Sigurd B. Angenent and Dan Knopf

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact

  8. On a dichotomy of the curvature decay of steady Ricci soliton

    Pak-Yeung Chan and Bo Zhu

    Topics Ricci-flat & Calabi–Yau, Boundaries & non-compact, Ricci solitons · Shared terms “complete noncompact”

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) Ricci solitons Metrics that change under Ricci flow only by scaling and diffeomorphisms: shrinking, steady or expanding solitons. Start with: The Ricci flow on surfaces (Hamilton, 1988) Ricci-flat & Calabi–Yau Ricci-flat metrics do not move under Ricci flow. Every known compact simply connected example has special holonomy (Calabi–Yau, hyperkähler, G₂… Start with: On the ricci curvature of a compact kähler manifold and the complex monge‐ampére equation, I (Yau, 1978) 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)

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.

  3. 11B.1
    Ricci SolitonsBook 11B · Solitons, Compactness and Singularities

    Self-similar solutions: the Gaussian, cylinders, the cigar and the Bryant soliton.