Analysis Seminar

We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons.

Our approach relies on two new techniques. The first, called the PDE-ODI principle, converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This framework bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. The second combines a new leading mode condition combined with an "induction over thresholds" argument" to obtain even finer asymptotic estimates.

This is joint work with Yi Lai.