Rotationally symmetric Ricci flow on ℝⁿ⁺¹

Francesco Di Giovanni

Paper · Apr 21, 2019

Rotationally symmetric Ricci flow on $R^{n + 1}$

Francesco Di Giovanni

math.DGarXiv:1904.09555v3

View PDF View HTML arXiv page

As on arXiv

Abstract

We study the Ricci flow on $R^{n + 1}$ , with $n \geq 2$ , starting at some complete bounded curvature rotationally symmetric metric $g_{0}$ . We first focus on the case where $(R^{n + 1}, g_{0})$ does not contain minimal hyperspheres; we prove that if $g_{0}$ is asymptotic to a cylinder then the solution develops a Type-II singularity and converges to the Bryant soliton, while if the curvature of $g_{0}$ decays at infinity, then the solution is immortal. As a corollary, we prove a conjecture by Chow and Tian about Perelman's standard solutions. We then consider a class of asymptotically flat initial data $(R^{n + 1}, g_{0})$ containing a neck and we prove that if the neck is sufficiently pinched, in a precise way, the Ricci flow encounters a Type-I singularity.

View PDF View HTML arXiv page

Submitted: Apr 21, 2019
Last revised: Feb 17, 2021 (v3)
Subjects: math.DG
Journal: Volume 381, 2021
DOI: 10.1016/j.aim.2021.107621

Comments: 26 pages, final version. Accepted in Adv. Math

Ricci flow Ancient solutions Singularities Boundaries & non-compact Long-time existence & convergence Rotational 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 2016, plus the canon.

Closer to the centre: more closely connected. Clockwise from the top: oldest to newest. Ricci flowWider flowsalso connected to each other

Ricci flow of warped Berger metrics on $R^{4}$
Francesco Di Giovanni2019

Topics Rotational symmetry & warped products, Ancient solutions, Long-time existence & convergence · Same author Francesco Di Giovanni · Shared terms “rotationally symmetric”, “solution develops”, “Bryant soliton”
Rotationally symmetric Ricci Flow on $R^{n + 1}$
Ming Hsiao2025

Topics Rotational symmetry & warped products, Singularities · Shared terms “symmetric Ricci”, “rotationally symmetric”, “minimal hypersphere”
Rotational symmetry of ancient solutions to the Ricci flow in dimension $3$ – The compact case
S. Brendle2019

Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Shared terms “rotationally symmetric”, “Bryant soliton”
An Anisotropic Entropy for the Rotationally Symmetric Ricci Flow
Hongxin Guo and Guozhi Lin2026

Topics Rotational symmetry & warped products, Singularities · Shared terms “symmetric Ricci”, “rotationally symmetric”
Convergence of Ricci flow solutions to Taub-NUT
Francesco Di Giovanni2020

Topics Long-time existence & convergence, Boundaries & non-compact, Singularities · Same author Francesco Di Giovanni · Shared terms “hyperspheres”, “asymptotically flat”
The three-dimensional standard solution to the Ricci flow is modeled by the Bryant soliton
Yongjia Zhang2017

Topics Ancient solutions · Shared terms “standard solutions”, “Bryant soliton”
Ancient solutions to the Ricci flow in higher dimensions
Xiaolong Li and Yongjia Zhang2018

Topics Rotational symmetry & warped products, Ancient solutions, Boundaries & non-compact · Shared terms “rotationally symmetric”, “Bryant soliton”
Ancient solutions to the Ricci flow in dimension 3
S. Brendle2018

Topics Ancient solutions, Boundaries & non-compact, Singularities · Shared terms “Bryant soliton”, “Perelman”

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) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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)

In the guidebook

12B.3
The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery
Necks and caps, compactness, and universal estimates.
11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.
11B.6
Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

[1] 12B.3
The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery
Necks and caps, compactness, and universal estimates.

[2] 11B.4
Blow-Ups, Blow-Downs and Splitting at InfinityBook 11B · Solitons, Compactness and Singularities
Limits of rescaled manifolds and flows: why blow-up limits are nonnegatively curved, what blow-downs see, and how lines appear at infinity.

[3] 11B.6
Rotationally Symmetric Flows and the Standard SolutionBook 11B · Solitons, Compactness and Singularities
The neckpinch in equations: warped products under the flow, what the maximum principle forces, and the standard solution that surgery glues in.

Rotationally symmetric Ricci flow on $R^{n + 1}$

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters

Rotationally symmetric Ricci flow on Rn+1

In the guidebook

Rotationally symmetric Ricci flow on $R^{n + 1}$