Optimal regularity for the arrival time equation

For a convex mean curvature flow in ${\mathbb{R}}^{n+1}$, the corresponding arrival time function solves a degenerate elliptic equation that becomes singular at extinction. We prove that it is $C^{24,\alpha }$ for any $0<\alpha <1$ in the plane, with a logarithmic modulus for its twenty-fourth derivatives. We prove that this bound is sharp combining regularity obstruction in our companion paper. We obtain asymptotic expansions of the rescaled flow to any prescribed order, including derivative estimates for the remainder. Combined with the work of Sesum for $n\ge 2$, this also gives the sharp $C^{2,2\mathrm{/}n}$ regularity for convex arrival time functions in higher dimensions.