The linearized translator equation and applications

In this paper, we consider the linearized translator equation L_ϕu=f, around entire convex translators M=graph (ϕ) ⁴, i. e. in the first dimension where the Bernstein property fails. Here, L_ϕu=div (a_ϕD u) + b_ϕ 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 L_ϕ 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 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.