Classification of quadratically pinched translators in higher codimension
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 (ℝn+p,⟨·,·⟩) satisfies |B|2≤ (4/3n−ε0)|H|2 for some ε0>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≤ (cn−ε0)|H|2, cn=min{4/3n,1/(n−2)} for n≥3, with cn=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.
Related
- Rigidity of Ricci-Pinched Complete Self-Shrinkers in Arbitrary Codimension
- Rigidity of Minimal Submanifolds in Spheres of Higher Codimension
- Tensorial pinching by deforming curvature cones in higher codimension mean curvature flow
- Ergodicity as an obstruction to classification by higher-rank invariants
- Ricci-pinched 3-Manifolds with second-fundamental-form-pinched Boundary
BibTeX
@misc{impera2026classification,
title = {{Classification of quadratically pinched translators in higher codimension}},
author = {Debora Impera and Michele Rimoldi and Francesco Ruatta},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01713},
eprint = {2610.01713},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01713},
}
RIS
TY - UNPB
TI - Classification of quadratically pinched translators in higher codimension
AU - Debora Impera
AU - Michele Rimoldi
AU - Francesco Ruatta
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01713
UR - https://arxiv.org/abs/2610.01713
AB - 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,⟨·,·⟩) satisfies |B|^2≤ (4/3n−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≤ (c_n−ε_0)|H|^2, c_n=min{4/3n,1/(n−2)} for n≥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.
ER -