2025/09/08 by Kyeongsu Choi, Choi, Kyeongsu, Robert Haslhofer +3 · 1 citation
Engineering · #Advanced Control Systems Design
paper · pdf · doi:10.48550/arxiv.2509.06667
In this paper, we consider the linearized translator equation Lϕu=f, around entire convex translators M=\textrmgraph(ϕ)⊂ℝ4, i.e. in the first dimension where the Bernstein property fails. Here, Lϕu=div (aϕD u)+ bϕ⋅ Du is a mean curvature type elliptic operator, whose coefficients degenerate as the slope tends to infinity. We derive two fundamental barrier estimates, specifically an upper-lower estimate and an inner-outer estimate, which allow to propagate L^∞-control between different regions. Packaging these and further estimates together we then develop a Fredholm theory for Lϕ between carefully designed weighted function spaces. Combined with Lyapunov-Schmidt reduction we infer that the space S of noncollapsed translators in ℝ4 is a finite dimensional analytic variety and that the tip-curvature map κ:S→ℝ is analytic. Together with the main result from our prior paper (Camb. J. Math. '23) this allows us to complete the classification of noncollapsed translators in ℝ4. In particular, we conclude that the one-parameter family of translators constructed by Hoffman-Ilmanen-Martin-White is uniquely determined by the smallest principal curvature at the tip.