Paper · Apr 22, 2026

A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary.II

Gang Li

math.DGarXiv:2604.20383v2
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Abstract

This is a continuation of the research in [16]. Let be a closed geodesic -ball in the hyperbolic space . Let be a positive constant. In this paper, we show that for , starting from the metric on , with certain prescribed non-decreasing rotationally symmetric mean curvature and the fixed conformal class on the boundary , the solution to the normalized Ricci flow which is continuous up to the boundary, exists for all , and converges locally uniformly in the interior of to a complete hyperbolic metric as (see Theorem 1.1 for details). Under some additional conditions, we show the same conclusion holds for .

Comments: Typos are corrected so that it is more readable

Ricci flowBoundaries & non-compactRotational symmetry & warped products

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  1. A Loewner-Nirenberg phenomena for Ricci flow on compact manifolds with boundary

    Gang Li PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact · Same author Gang Li · Shared terms “loewner nirenberg”, “nirenberg phenomena”, “decreasing rotationally”

  2. A normalized Ricci flow on surfaces with boundary towards the complete hyperbolic metric

    Gang Li PDF

    Topics Ricci flow · Same author Gang Li · Shared terms “complete hyperbolic”, “hyperbolic metric”, “normalized Ricci”

  3. Gradient estimates for a parabolic partial differential equation under the Ricci-Bourguignon flow

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    Topics Rotational symmetry & warped products, Boundaries & non-compact

  4. Convergence stability for Ricci flow on manifolds with bounded geometry

    Eric Bahuaud, Christine Guenther, James Isenberg et al. PDF

    Topics Rotational symmetry & warped products, Boundaries & non-compact · Shared terms “hyperbolic metric”, “rotationally symmetric”

  5. Exponential Asymptotics of Ricci Flow Neckpinch

    Hamidreza Mahmoudian PDF

    Topics Rotational symmetry & warped products · Shared terms “compact manifolds”, “rotationally symmetric”

  6. Existence of singular rotationally symmetric gradient Ricci solitons in higher dimensions

    Kin Ming Hui PDF

    Topics Rotational symmetry & warped products · Shared terms “rotationally symmetric”

  7. Ricci flow of discrete surfaces of revolution, and relation to constant Gaussian curvature

    Naoya Suda PDF

    Topics Rotational symmetry & warped products

  8. Symmetries of Ricci Flows

    Enrique López, Stylianos Dimas and Yuri Bozhkov PDF

    Topics Rotational symmetry & warped products

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) Boundaries & non-compact Ricci flow beyond closed manifolds: complete non-compact spaces, manifolds with boundary, and asymptotically flat or hyperbolic ends. Start with: Deforming the metric on complete Riemannian manifolds (Shi, 1989) Rotational symmetry & warped products For rotationally symmetric metrics and warped products, Ricci flow reduces to a parabolic equation in one space variable. Start with: The ricci flow on radially symmetric R3 (Ivey, 1994)

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.