Abstract:

We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges at a super-exponential rate, then it must coincide with the cylinder itself. We also show that this result is sharp by constructing local graphical counterexamples with arbitrarily fast super-exponential convergence and rapidly expanding domains. These examples form infinite-dimensional families of Tikhonov-type solutions and show that unique continuation fails for local graphical solutions. Our constructions apply to a broad class of nonlinear equations. This talk is based on joint work with Yiqi Huang.