Arnold-Thom conjecture for the arrival time of surfaces

Following ojasiewicz's uniqueness theorem and Thom's gradient conjecture, Arnold proposed a stronger version about the existence of limit tangents of gradient flow lines for analytic functions. We prove ojasiewicz's theorem and Arnold's conjecture in the context of arrival time functions for mean curvature flows in Rⁿ⁺¹ with neck or non-degenerate cylindrical singularities. In particular, we prove the conjectures for all mean convex mean curvature flows of surfaces, including the cases when the arrival time functions are not C². The results also apply to mean curvature flows starting from two-spheres or generic closed surfaces.