A delayed interior area-to-height estimate for the Curve Shortening Flow

TL;DR

This work develops a delayed parabolic regularity framework for the Graphical Curve Shortening Flow, introducing a magic time $t_* = \mathcal{A}/\pi$ that governs when interior height control becomes available. It proves a delayed $L^1_{loc}$ to $L^{\infty}_{loc}$ height estimate, from which Evans–Spruck type instantaneous gradient bounds follow, yielding an $L^{p>1}_{loc}$ to $L^{\infty}_{loc}$ estimate. The paper further extends the existence theory by constructing graphical CSFs starting from non-atomic Radon measures $\nu \in \mathcal{M}_*(\mathbb{R})$, with weak convergence to $\nu$ and strong convergence away from singularities, and develops a local Harnack quantity to underpin the interior estimates. Together, these results broaden graphical CSF theory to measure-valued initial data and clarify the delayed regularization mechanism, connecting to analogous Ricci-flow methods. The work also discusses a correspondence with the Chou–Kwong framework and lays groundwork for a broader dominated-flow perspective in 1D CSF.

Abstract

In (Sobnack & Topping; 2024a, 2024b), Topping & the author proposed the principle of 'delayed parabolic regularity' for the Curve Shortening Flow; in (Sobnack & Topping, 2024a), they provided a handful of proper graphical situations in which their delayed regularity framework is valid. In this paper, we show that there holds an interior graphical estimate for the Curve Shortening Flow in the spirit of the proposed framework. More precisely, we show the following $ \mathrm{L}^1_\mathrm{loc} $-to-$ \mathrm{L}^\infty_\mathrm{loc} $ estimate: If a smooth Graphical Curve Shortening Flow $ u : (-1, 1) \times [0, T) \mapsto \mathbb{R} $ starts from a function $ u_0 := u( \, \cdot \, , 0) : (-1, 1) \mapsto \mathbb{R} $ with $ \mathrm{L}^1(\!(-1,1)\!) $ norm strictly less than $ πT $, then after waiting for the 'magic time' $ t_\star : = \| u_0 \|_{\mathrm{L}^1(\!(-1,1)\!)} / π$, the size $ |u(0, t)| $ of $ u( \, \cdot \, , t) $ at the origin at any time $ t \in (t_\star, T) $ is controlled purely in terms of $ \| u_0 \|_{\mathrm{L}^1(\!(-1,1)\!)} $ and $ t - t_\star $. We apply our estimate to construct Graphical Curve Shortening Flows starting weakly from Radon measures without point masses.

Figures (4)

• Figure 2.1: Construction of $\gamma_0 : \mathbb{R} \mapsto \mathbb{R}^2$ from a given $u_0 \in \mathrm{C}^{0,1}(J_1; \mathbb{R}_{\geq 0})$.
• Figure 2.2: Left: Applying the Avoidance Principle (Theorem \ref{['thm:avoidii']}) to the solid $\gamma$ and to a dashed Angenent oval solution $\mathfrak{o}$ in \ref{['num:circ']}. Right: Applying Angenent's Intersection Principle (Theorem \ref{['thm:inter']}) to the solid $\gamma$ and to a dashed vertical/horizontal line $V_r$/$H_{r'}$ in \ref{['num:line']}.
• Figure 2.3: Illustration of $\mathcal{A}(y, t)$ and $\phi(y, t)$.
• Figure 3.1: The shaded region trapped below $\gamma_u(t)$ must contain the hatched trapezium, and so in particular the area of the hatched trapezium must be bounded from above by the area $\mathcal{A}(c_t, t) = \mathcal{A}(1, t)/2 = \overline{\mathcal{A}}/2 + \pi t /2$ of the shaded region.

Theorems & Definitions (36)

• Theorem 1.1: Delayed $\mathrm{L}^1$-to-$\mathrm{L}^\infty$ estimate; sobtopdelayed
• Theorem 1.5: Variant of sobtopdelayed; see sobphd
• Theorem 1.8: Delayed $\mathrm{L}^1_\mathrm{loc}$-to-$\mathrm{L}^\infty_\mathrm{loc}$ estimate
• Theorem 1.11: Instantaneous $\mathrm{L}^{p>1}_\mathrm{loc}$-to-$\mathrm{L}^\infty_\mathrm{loc}$ estimate; cf. chou2020general
• Theorem 1.13: Existence from $\mathcal{M}_*(\mathbb{R})$--initial data
• Lemma 2.1
• proof
• Remark 2.4
• Definition 2.6: Local GCSF
• Definition 2.7: Local Harnack quantity; cf. sobtopdelayed