A delayed interior area-to-height estimate for the curve shortening flow

The principle of delayed parabolic regularity for the Curve Shortening Flow—that if two evolving curves bound a region of area $ \mathcal{{A}} $, then, starting from time $ \mathcal{{A}}/\pi $, the regularity of one curve is controllable in terms of the time elapsed, the area $ \mathcal{{A}} $ and the regularity of the other curve—was proposed by Topping & the author in (Sobnack & Topping, 2024), where they also provided a number of graphical situations in which their delayed regularity framework is valid. In this paper, we generalise some of the results in (Sobnack & Topping, 2024) within the graphical setting, ultimately by showing that there holds an interior graphical estimate for the Curve Shortening Flow in the spirit of the proposed framework. We also provide a few applications of our estimate, such as the existence of Graphical Curve Shortening Flows starting weakly from Radon measures without point masses.