Generic Regularity of Level Set Flows with Spherical Singularity

The paper proves that for n ≥ 2 there is an open dense subset of mean-convex initial hypersurfaces with a single spherical singularity for which the level-set arrival time is C2 but not C3 (Theorem 1.1), by showing that generically the rescaled flow converges to the sphere at the slow rate governed by the third eigenfunctions of L=Δ+1 (i.e., the degree-2 spherical harmonics with eigenvalue −1/n), and invoking Strehlke’s asymptotics to force failure of C3 at the maximum . The candidate solution misidentifies the slow mode as “l=3” (degree 3) rather than the paper’s “third eigenfunctions” (degree 2), contradicting the spectrum listed in the paper (λ0=1, λ1=1/2, λ2=−1/n) . It also overstates a global C2 regularity claim for all mean-convex flows that is neither established nor needed by the paper, which instead relies on Strehlke’s expansion for the C3 obstruction and its own dynamical argument for genericity .