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arXiv · 2305.19137

Enhanced profile estimates for ovals and translators

Kyeongsu Choi·Robert Haslhofer·Or Hershkovits
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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 $\mathbb{R}^4$, we derive these estimates in the context of ancient ovals in $\mathbb{R}^3$ and noncollapsed translators in $\mathbb{R}^4$, though our methods seem to apply in other settings as well.

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Kyeongsu Choi, Robert Haslhofer, Or Hershkovits. 2023-05-30. Enhanced profile estimates for ovals and translators. https://arxiv.org/abs/2305.19137

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