Paper · Oct 1, 2026

Classification of quadratically pinched translators in higher codimension

Debora Impera, Michele Rimoldi, Francesco Ruatta

As on arXiv

Abstract

We study complete translating solitons for the mean curvature flow in arbitrary codimension under quadratic pinching of the second fundamental form. We prove a codimension reduction theorem at the Andrews-Baker threshold: if a complete $n$ -dimensional connected translator isometrically immersed in $(R^{n + p}, ⟨ \cdot, \cdot ⟩)$ satisfies $∣ B ∣^{2} \leq (\frac{4}{3 n} - ε_{0}) ∣ H ∣^{2}$ for some $ε_{0} > 0$ , then either it is an affine $n$ -plane or $∣ H ∣ > 0$ everywhere and the translator is contained in an $(n + 1)$ -dimensional affine subspace. In particular, this improves in low dimensions the pinching range previously obtained from codimension reduction results for ancient mean curvature flows. Our proof is purely elliptic: viewing translators as weighted minimal submanifolds, we combine Simons-type identities for the drift Laplacian with refined gradient estimates and an Omori-Yau maximum principle. As a consequence, under the stronger pinching condition $∣ B ∣^{2} \leq (c_{n} - ε_{0}) ∣ H ∣^{2}, c_{n} = min {\frac{4}{3 n}, \frac{1}{n - 2}}$ for $n \geq 3$ , with $c_{n} = 2/3$ when $n = 2$ , we obtain a rigidity theorem: the translator is either an affine plane or a bowl soliton contained in an $(n + 1)$ -dimensional affine subspace. No entropy assumption is required.

Comments: 26 pages. Comments are welcome!

Mean curvature flow Ancient solutions Self-shrinkers & translators Curvature pinching Stability & uniqueness Minimal & CMC surfaces

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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. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Ancient solutions Solutions that exist for all negative times. Every blow-up of a singularity is one, so classifying ancient solutions means classifying the shapes… Start with: The entropy formula for the Ricci flow and its geometric applications (Perelman, 2002) 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) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982)

In the guidebook

6A.9
Reaction–Diffusion: Fronts, Patterns and PinchingBook 6A · The Heat Equation and Its Relatives
Critical patch size, travelling fronts and their speeds, bistable waves and the Allen–Cahn equation, Turing patterns, and the curvature of a Ricci flow as a reaction–diffusion system.
11A.7
Maximum Principles under Ricci FlowBook 11A · The Ricci Flow: Existence and Maximum Principles
Scalar and tensor maximum principles, and preserved curvature conditions.
11A.10
Hamilton–Ivey PinchingBook 11A · The Ricci Flow: Existence and Maximum Principles
Why high curvature in dimension 3 is almost positive.

Classification of quadratically pinched translators in higher codimension

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