The linearized translator equation and applications

Abstract

In this paper, we consider the linearized translator equation $L_\phi u=f$, around entire convex translators $M=\textrm{graph}(\phi)\subset\mathbb{R}^4$, i.e. in the first dimension where the Bernstein property fails. Here, $L_\phi u=\mathrm{div} (a_\phi D u)+ b_\phi\cdot Du$ 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 $L^\infty$-control between different regions. Packaging these and further estimates together we then develop a Fredholm theory for $L_\phi$ between carefully designed weighted function spaces. Combined with Lyapunov-Schmidt reduction we infer that the space $\mathcal{S}$ of noncollapsed translators in $\mathbb{R}^4$ is a finite dimensional analytic variety and that the tip-curvature map $\kappa:\mathcal{S}\to\mathbb{R}$ 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 $\mathbb{R}^4$. 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.