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Abstract: In this paper, we develop some analytic foundations for the linearized translator equation in R4, 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, namely for the derivative of the variation field Win 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, namely the derivative of the profile function Yin the tip region. Moreover, together with interior estimates by virtue of the linearized translator equation our gradient estimate implies a bound for W tau as well. Hence, our gradient estimate also serves as substitute for Hamilton's Harnack inequality, which has played an important role for controlling Y tau in the tip region.
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