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- Title
- Enhanced profile estimates for ovals and translators
- KIAS Author
- Choi, Kyeongsu
- Journal
- ADVANCES IN MATHEMATICS, 2024
- Archive
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- 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 C 0-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 C 0-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 R 4 , we derive these estimates in the context of ancient ovals in R 3 and noncollapsed translators in R 4 , though our methods seem to apply in other settings as well. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.