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Ancient curve shortening flow in the disc with mixed boundary condition
Mat Langford, Yuxing Liu, George McNamara
Given any non-central interior point of the unit disc , the diameter through is the union of two linear arcs emanating from which meet orthogonally,…
Sharp distance comparison for curve shortening flow on the round sphere
Paul Bryan, Mat Langford, Jonathan J. Zhu
We prove that curve shortening flow on the round sphere displays sharp chord-arc improvement, precisely as in the planar setting (Andrews and Bryan, Comm. Anal. Geom., 2011). As in…
Ancient pancake solutions to fully nonlinear curvature flows
Sathya Rengaswami, Mat Langford
We construct -invariant ancient ``pancake'' solutions to a large and natural class of fully nonlinear curvature flows. We then establish that these are the unique…
Sharp one-sided curvature estimates for mean curvature flow and applications
Mat Langford
We prove a sharp pinching estimate for immersed mean convex solutions of mean curvature flow which unifies and improves all previously known pinching estimates, including the umbil…