arXiv subjects

Arjun Sobnack

Publications and source records attributed to Arjun Sobnack.

4 recordsLinked to original sources

arXiv2026

The Harnack inequality without convexity for curve shortening flow

In 1995, Hamilton introduced a Harnack inequality for convex solutions of the mean curvature flow. In this paper we prove an alternative Harnack inequality for curve shortening flow, i.e. one-dimensional mean curvature flow, that does not require any assumption of convexity. For an initial proper curve in the plane whose ends are radial lines but which is otherwise arbitrarily wild, we use the Harnack inequality to give an explicit time by which the curve shortening flow evolution must become graphical. This gives a new instance of delayed parabolic regularity. The Harnack inequality also gives estimates describing how a polar graphical flow with radial ends settles down to an expanding solution. Finally, we relate our Harnack inequality to Hamilton's by identifying a pointwise curvature estimate implied by both Harnack inequalities in the special case of convex flows.

math.DG↗

arXiv2025

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.

math.AP↗

arXiv2024

Monotonicity of the modulus under curve shortening flow

Given two disjoint nested embedded closed curves in the plane, both evolving under curve shortening flow, we show that the modulus of the enclosed annulus is monotonically increasing in time. An analogous result holds within any ambient surface satisfying a lower curvature bound.

math.DG↗

arXiv2024

Delayed parabolic regularity for curve shortening flow

Given two curves bounding a region of area $A$ that evolve under curve shortening flow, we propose the principle that the regularity of one should be controllable in terms of the regularity of the other, starting from time $A/\pi$. We prove several results of this form and demonstrate that no estimate can hold before that time. As an example application, we construct solutions to graphical curve shortening flow starting with initial data that is merely an $L^1$ function.

math.AP↗