Uniqueness of convex ancient solutions to mean curvature flow in ℝ³

S. Brendle; K. Choi

Paper · Nov 2, 2017

Uniqueness of convex ancient solutions to mean curvature flow in $R^{3}$

S. Brendle, K. Choi

math.DGarXiv:1711.00823v3

View PDF View HTML arXiv page

As on arXiv

Abstract

A well-known question of Perelman concerns the classification of noncompact ancient solutions to the Ricci flow in dimension $3$ which have positive sectional curvature and are $κ$ -noncollapsed. In this paper, we solve the analogous problem for mean curvature flow in $R^{3}$ , and prove that the rotationally symmetric bowl soliton is the only noncompact ancient solution of mean curvature flow in $R^{3}$ which is strictly convex and noncollapsed.

View PDF View HTML arXiv page

Submitted: Nov 2, 2017
Last revised: Jan 14, 2019 (v3)
Subjects: math.DG, math.AP

Comments: revised version, to appear in Invent. Math

Ricci flow Mean curvature flow Ancient solutions Self-shrinkers & translators (Non)collapsing Sphere theorems Stability & uniqueness Boundaries & non-compact 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

Uniqueness of convex ancient solutions to mean curvature flow in higher dimensions
S. Brendle and K. Choi2018Wider flows

Topics Rotational symmetry & warped products, (Non)collapsing, Ancient solutions · Same authors S. Brendle, K. Choi · Shared terms “convex ancient”, “noncompact ancient”, “noncollapsed”
Rotational symmetry of ancient solutions to the Ricci flow in higher dimensions
Simon Brendle and Keaton Naff2020

Topics Sphere theorems, Rotational symmetry & warped products, (Non)collapsing · Same author S. Brendle · Shared terms “noncompact ancient”, “noncollapsed”
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 · Same author S. Brendle · Shared terms “noncompact ancient”, “rotationally symmetric”
Ancient solutions to the Ricci flow in dimension 3
S. Brendle2018

Topics Ancient solutions, Boundaries & non-compact · Same author S. Brendle · Shared terms “noncompact ancient”, “Perelman”
Uniqueness of compact ancient solutions to three-dimensional Ricci flow
Simon Brendle, Panagiota Daskalopoulos and Natasa Sesum2020

Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same author S. Brendle · Shared terms “noncollapsed”, “Perelman”
Uniqueness of compact ancient solutions to the higher dimensional Ricci flow
Simon Brendle, Panagiota Daskalopoulos, Keaton Naff et al.2021

Topics (Non)collapsing, Ancient solutions, Stability & uniqueness · Same author S. Brendle · Shared terms “noncollapsed”, “Perelman”
Ancient solutions to the Ricci flow in higher dimensions
Xiaolong Li and Yongjia Zhang2018

Topics Sphere theorems, Rotational symmetry & warped products, (Non)collapsing · Shared terms “noncollapsed”, “rotationally symmetric”
Enhanced profile estimates for ovals and translators
Kyeongsu Choi, Robert Haslhofer and Or Hershkovits2023

Topics (Non)collapsing, Ancient solutions, Self-shrinkers & translators · Same author K. Choi · Shared terms “convex ancient”, “noncollapsed”

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.

Foundational paper in the Ancient solutions guide: Brendle–Choi: the mean curvature flow counterpart in $R^{3}$ .

Foundational paper in the Mean curvature flow guide: Brendle–Choi: the bowl is the only noncompact convex noncollapsed ancient solution in $R^{3}$ .

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) Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) 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) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990)

In the guidebook

12B.4
Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.
11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.
9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

[1] 12B.4
Compactness and Universal Estimates for κ-SolutionsBook 12B · κ-Solutions and Surgery
Asymptotic volume zero, curvature controlled by volume, Perelman's Theorem 11.7, a universal κ, neck-like behaviour at infinity, and the constants that canonical neighbourhoods inherit.

[2] 11A.11
Hamilton’s 1982 TheoremBook 11A · The Ricci Flow: Existence and Maximum Principles
Positive Ricci curvature in dimension 3 flows to a round metric.

[3] 9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

Uniqueness of convex ancient solutions to mean curvature flow in $R^{3}$

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters

Uniqueness of convex ancient solutions to mean curvature flow in R3

In the guidebook

Uniqueness of convex ancient solutions to mean curvature flow in $R^{3}$