Geometric Flows have been one of the most fruitful research area in geometric analysis the past decades with vast applications such as the resolution of the Geometrization conjecture, the Riemannian Penrose conjecture and the differentiable sphere theorem. Hereby the main goal is often to precisely understand the long time behavior of the flow. In1996 Stahl studied free boundary mean curvature flow whose barrier is given by a round sphere and showed that convex surfaces converge to shrinking hemispheres along the flow. This can be compared to Huisken's celebrated convergence result for mean curvature flow. An alternative approach to Stahl's result has also been given by Edelen in 2016. This raises the natural question whether the assumption of the barrier being a round sphere can be removed and that a similar result holds true for arbitrary convex barrier surfaces. Using a novel 5-tensor perturbation technique we confirm this in the 2-dimensional setting.

This work is joint with Martin Li.