Rotational symmetry of self-similar solutions to the Ricci flow

By S. Brendle. 2012

175 citations recorded by the source corpus.

Let (M,g) be a three-dimensional steady gradient Ricci soliton which is non-flat and κ-noncollapsed. We prove that (M,g) is isometric to the Bryant soliton up to scaling. This solves a problem mentioned in Perelman’s first paper.

Subfields: Applied Mathematics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ad45ea63a80f58e0ccbd43bdccaeeb5b89a1b347

How researchers see this paper

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In Applied Mathematics

1. Robert Haslhofer
2. Lu Wang
3. S. Brendle
4. Brett Kotschwar
5. G. Catino
6. L. Mazzieri
7. G. Huisken
8. Qiang Chen

In Geometry and Topology

1. Lu Wang
2. Robert Haslhofer
3. G. Catino
4. S. Brendle
5. Brett Kotschwar
6. L. Mazzieri
7. G. Huisken
8. Qiang Chen

Citations in the corpus

• Bach-flat gradient steady Ricci solitons (2011, background): “More recently, Brendle [3] (see also Proposition 5.2 below) showed that for an 3-dimensional gradient steady soliton ( M 3 , g ij , f ) if the scalar curvature R is positive and tends to zero at infinity, and that ( M 3 , g ij , f ) is asymptotic to the Bryant soliton in some suitable sense, then ( M”
• Rigidity of Complete Gradient Steady Ricci Solitons with Harmonic Weyl Curvature (2021, unknown): “In [3], Brendle showed that the Bryant soliton is the only complete noncompact, nonflat, κ-noncollapsed, gradient steady Ricci soliton, proving a conjecture by Perelman [36].”
• On the geometry of gradient einstein-type manifolds (2014, unknown): “[25], [28], [39], [22], [38], [48], [37], [40], [10], [7], [14], [42], [11], [13], [8], [6], [33], [5] and references therein).”
• Gradient Einstein solitons (2012, unknown): “Brendle posted the article [3] on the ArXiv, where Perelman’s claim is proved.”
• Uniqueness of the bowl soliton (2014, methodology): “Uniform 2-convexity is preserved under mean curvature flow, and arises e.g. in the construction of mean curvature flow with surgery by Huisken-Sinestrari [HS09], see also Haslhofer-Kleiner [HK14] and Brendle-Huisken [BH].”
• Embedded self-similar shrinkers of genus 0 (2014, unknown): “[4], [5]).”
• Ancient solutions to the Ricci flow in dimension $3$ (2018, methodology): “By [6], the limit must be the rotationally symmetric Bryant soliton.”
• Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, methodology): “Brendle’s approach in [6] combines the construction of “approximate Killing vector fields” with a careful blow-down analysis and a Liouville-type theorem for solutions to the Lichnerowicz PDE.”
• Mean curvature flow with surgery of mean convex surfaces in R3\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb (2013, methodology): “Finally, the first author has recently obtained a classification of self-similar solutions to the Ricci flow in dimension 3 under a noncollapsing assumption (see [3]).”