Paper · Nov 4, 2022

Spectral quantization for ancient asymptotically cylindrical flows

As on arXiv

Abstract

We study ancient mean curvature flows in $R^{n + 1}$ whose tangent flow at $- \infty$ is a shrinking cylinder $R^{k} \times S^{n - k} (2 (n - k) ∣ t ∣)$ , where $1 \leq k \leq n - 1$ . We prove that the cylindrical profile function $u$ of these flows have the asymptotics $u (y, ω, τ) = (y^{⊤} Q y - 2 tr (Q)) /∣ τ ∣ + o (∣ τ ∣^{- 1})$ as $τ \to - \infty$ , where the cylindrical matrix $Q$ is a constant symmetric $k \times k$ matrix whose eigenvalues are quantized to be either 0 or $- \frac{2 ( n - k )}{4}$ . Compared with the bubble-sheet quantization theorem in $R^{4}$ obtained by Haslhofer and the first author, this theorem has full generality in the sense of removing noncollapsing condition and being valid for all dimensions. In addition, we establish symmetry improvement theorem which generalizes the corresponding results of Brendle-Choi and the second author to all dimensions. Finally, we give some geometric applications of the two theorems. In particular, we obtain the asymptotics, compactness and $O (n - k + 1)$ symmetry of $k$ -ovals in $R^{n + 1}$ which are ancient noncollapsed flows in $R^{n + 1}$ satisfying full rank condition that $rk (Q) = k$ , and we also obtain the classification of ancient noncollapsed flows in $R^{n + 1}$ satisfying vanishing rank condition that $rk (Q) = 0$ .

Comments: 58 pages

Mean curvature flow Ancient solutions Singularities (Non)collapsing Limit spaces & RCD Eigenvalues & spectrum

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.

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

Rigidity of ancient ovals in higher dimensional mean curvature flow
Beomjun Choi, Wenkui Du and Jingze Zhu2025Wider flows PDF

Topics Eigenvalues & spectrum, (Non)collapsing, Ancient solutions · Same authors Wenkui Du, Jingze Zhu · Shared terms “ancient noncollapsed”, “cylindrical matrix”, “full rank”
Hearing the shape of ancient noncollapsed flows in $R^{4}$
Wenkui Du and Robert Haslhofer2021Wider flows PDF

Topics Eigenvalues & spectrum, (Non)collapsing, Ancient solutions · Same author Wenkui Du · Shared terms “noncollapsed flows”, “ancient noncollapsed”, “bubble sheet”
Classification of ancient ovals in higher dimensional mean curvature flow
Beomjun Choi, Wenkui Du and Ziyi Zhao2026Wider flows PDF

Topics Eigenvalues & spectrum, (Non)collapsing, Ancient solutions · Same author Wenkui Du · Shared terms “ancient asymptotically”, “cylindrical flows”, “asymptotically cylindrical”
A nonexistence result for rotating mean curvature flows in $R^{4}$
Wenkui Du and Robert Haslhofer2022Wider flows PDF

Topics (Non)collapsing, Ancient solutions, Singularities · Same author Wenkui Du · Shared terms “noncollapsed flows”, “ancient noncollapsed”, “tangent flow”
Quantitative long range curvature estimate for mean curvature flow
Jingze Zhu2021Wider flows PDF

Topics (Non)collapsing, Ancient solutions · Same author Jingze Zhu · Shared terms “noncollapsed”
Classification of bubble-sheet ovals in $R^{4}$
Beomjun Choi, Panagiota Daskalopoulos, Wenkui Du et al.2022Wider flows PDF

Topics Ancient solutions · Same author Wenkui Du · Shared terms “bubble sheet”
Unique asymptotics of $S O (k) \times S O (n - k + 1)$ symmetric ancient ovals of Ricci flow
Panagiota Daskalopoulos, Wenkui Du, Natasa Sesum et al.2026 PDF

Topics Ancient solutions, Singularities · Same author Wenkui Du · Shared terms “asymptotics”, “profile function”, “tangent flow”
Classification of ancient noncollapsed flows in $R^{4}$
Kyeongsu Choi and Robert Haslhofer2024Wider flows PDF

Topics (Non)collapsing, Ancient solutions, Singularities · Shared terms “ancient noncollapsed”, “noncollapsed flows”, “bubble sheet”

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.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. 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) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) (Non)collapsing At scales where curvature is controlled, balls in a Ricci flow cannot be arbitrarily thin: Perelman's κ-noncollapsing theorem. Start with: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds (Cheeger et al., 1982)

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.
12B.3
The Structure of κ-SolutionsBook 12B · κ-Solutions and Surgery
Necks and caps, compactness, and universal estimates.
9B.3
Collapsing and NoncollapsingBook 9B · Comparison, Convergence and Heat on Manifolds
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.

Spectral quantization for ancient asymptotically cylindrical flows

Abstract

Connected papers

Background reading

In the guidebook

Keyboard shortcuts

Anywhere

Lists of papers

Guidebook chapters