As on arXiv
Abstract
We consider the profile function of ancient ovals and of noncollapsed translators. Recall that pioneering work of Angenent-Daskalopoulos-Sesum (JDG '19, Annals '20) gives a sharp -estimate and a quadratic concavity estimate for the profile function of two-convex ancient ovals, which are crucial in their papers as well as a slew of subsequent papers on ancient solutions of mean curvature flow and Ricci flow. In this paper, we derive a sharp gradient estimate, which enhances their -estimate, and a sharp Hessian estimate, which can be viewed as converse of their quadratic concavity estimate. Motivated by our forthcoming work on ancient noncollapsed flows in , we derive these estimates in the context of ancient ovals in and noncollapsed translators in , though our methods seem to apply in other settings as well.
Comments: 18 pages
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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.
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Classification of ancient noncollapsed flows in
Topics (Non)collapsing, Ancient solutions, Self-shrinkers & translators · Same authors Kyeongsu Choi, Robert Haslhofer · Shared terms “ancient noncollapsed”, “noncollapsed flows”, “Daskalopoulos Sesum”
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Ancient cylindrical flows with fast or slow convergence
Topics Ancient solutions · Same authors Kyeongsu Choi, Robert Haslhofer · Shared terms “ancient ovals”
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Hearing the shape of ancient noncollapsed flows in
Topics (Non)collapsing, Ancient solutions · Same author Robert Haslhofer · Shared terms “noncollapsed flows”, “ancient noncollapsed”, “ancient ovals”
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Rigidity of ancient ovals in higher dimensional mean curvature flow
Topics (Non)collapsing, Ancient solutions · Shared terms “ancient ovals”, “concavity estimate”, “quadratic concavity”
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A nonexistence result for rotating mean curvature flows in
Topics (Non)collapsing, Ancient solutions · Same author Robert Haslhofer · Shared terms “noncollapsed flows”, “ancient noncollapsed”
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Classification of bubble-sheet ovals in
Topics Ancient solutions · Same author Robert Haslhofer · Shared terms “ancient ovals”
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A note on blowup limits in 3d Ricci flow
Topics Ancient solutions · Same author Robert Haslhofer · Shared terms “ancient ovals”
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Classification of ancient ovals in higher dimensional mean curvature flow
Topics (Non)collapsing, Ancient solutions · Shared terms “ancient ovals”, “ancient noncollapsed”, “quadratic”
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.
In the guidebook
- 12B.4Compactness and Universal Estimates for κ-Solutions
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.
- 9B.3Collapsing and Noncollapsing
Injectivity radius bounds from volume, collapsing examples, and the definition Perelman needs.
- 12A.4κ-Noncollapsing
The theorem that excludes the cigar.