A gradient estimate for the linearized translator equation

In this paper, we develop some analytic foundations for the linearized translator equation in R4[jls-end-space/], i.e. in the first dimension where the Bernstein property fails. This equation governs how the (noncompact) singularity models of the mean curvature flow in R4 fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for Wv[jls-end-space/], namely for the derivative of the variation field W in the tip region. This serves as a substitute for the fundamental quadratic concavity estimate from Angenent-Daskalopoulos-Sesum, which has been crucial for controlling Yv[jls-end-space/], namely the derivative of the profile function Y in the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for Wτ as well. Hence, our gradient estimate also serves as substitute for Hamilton's Harnack inequality, which has played an important role for controlling Yτ in the tip region.