2025/09/09 by Choi, Kyeongsu, Haslhofer, Robert, Hershkovits, Or
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2509.07629
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 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, 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 Hamilton's Harnack inequality, which has played an important role for controlling Yτ in the tip region.