A gradient estimate for the linearized translator equation

Kyeongsu Choi; Robert Haslhofer; Or Hershkovits

Paper · Sep 9, 2025

A gradient estimate for the linearized translator equation

Kyeongsu Choi, Robert Haslhofer, Or Hershkovits

math.APWider flowsarXiv:2509.07629v2

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Abstract

In this paper, we develop some analytic foundations for the linearized translator equation in $R^{4}$ , 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 $R^{4}$ fit together in a common moduli space. Here, we prove a gradient estimate, which gives a sharp bound for $W_{v}$ , 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 $Y_{v}$ , 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 $W_{τ}$ as well. Hence, our gradient estimate also serves as substitute Hamilton's Harnack inequality, which has played an important role for controlling $Y_{τ}$ in the tip region.

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Submitted: Sep 9, 2025
Last revised: Mar 22, 2026 (v2)
Subjects: math.AP, math.DG

Comments: 20 pages (v2: minor corrections)

Mean curvature flow Self-shrinkers & translators Singularities Harnack inequalities Boundaries & non-compact

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Background reading

The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Self-shrinkers & translators Wider flows Self-similar solutions of mean curvature flow: shrinkers move by scaling, expanders by scaling outward, translators by sliding. Start with: Asymptotic behavior for singularities of the mean curvature flow (Huisken, 1990) Singularities On a closed manifold the flow can stop in finite time: curvature blows up, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) Harnack inequalities Differential Harnack inequalities compare a solution with itself at different points and times. Hamilton's matrix Harnack inequality says curvature… Start with: On the parabolic kernel of the Schrödinger operator (Li et al., 1986)

In the guidebook

6D.5
Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation
Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.
11B.2
Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities
Comparing curvature across space and time.
12A.6
𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing
Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.

[1] 6D.5
Diffusions on Manifolds and CurvatureBook 6D · Diffusions, Finance and Computation
Brownian motion on a Riemannian manifold, the Bakry–Émery calculus, curvature-dimension inequalities, gradient estimates and log-Sobolev, coupling, and Brownian motion along a Ricci flow.

[2] 11B.2
Ancient Solutions and the Harnack InequalityBook 11B · Solitons, Compactness and Singularities
Comparing curvature across space and time.

[3] 12A.6
𝓛-Geodesics and the Variational FormulasBook 12A · Entropy and Noncollapsing
Perelman's §7.1 derived line by line: the first and second variation of 𝓛-length, 𝓛-Jacobi fields, adapted fields, Hamilton's Harnack expressions, and the differential inequalities for the reduced distance.

A gradient estimate for the linearized translator equation

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