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Topping & the speaker proposed, in 2024, the principle that if two disjoint embedded (and sufficiently nice) Curve Shortening Flows (CSFs) bound an evolving connected region of fixed area A, then the regularity of one should be controlled after time A/π by the time, the area A and the regularity of the other. A handful of concrete results in the graphical setting which fit into the framework were provided.
In this talk I will discuss further work into this principle. Here we relax the grahicality constraint and ask the same question: Does the CSF regularise after a time depending on only the area? I will present a positive result in the flavour of this direction. Under a relaxed notion of ‘swept area’, given a (sufficiently nice) non-graphical curve, the CSF takes it to a graphical curve with controlled geometry after a time depending on only the swept area.
Time permitting, I will explain some aspects of the proof; one of which is an ‘alternative’ Harnack quantity/inequality for the CSF which does not require convexity.
All work is joint with P. M. Topping.