https://neckpinch.com/authors/francesco-ruatta/New arXiv papers by Francesco Ruatta, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2610.39877arxiv:2610.39877Thu, 29 Oct 2026 00:00:00 GMTFrancesco Ruatta — We derive an interior planarity estimate for space-time proper mean curvature flows of n-dimensional submanifolds of ℝ^(n+p), n≥5, that are uniformly quadratically pinched up to the Andrews-Baker threshold 4/3n. The estimate is parabolically scale-invariant and it requires neither curvature nor entropy bounds. The proof combines Naff's gradient estimates with a localization argument in the spirit of Ecker and Huisken. In dimensions five and six, Naff's estimates alone do not give the gradient terms a sign. We bound these terms from below with an argument we developed in a previous work. As a consequence, for n≥5, and by known results for 2≤ n≤4, every space-time proper ancient solution that is uniformly 4/3n-pinched is either a static affine n-plane or lies in a fixed affine (n+1)-dimensional subspace. This replaces the finite entropy and polynomial curvature growth assumptions of Lee, Naff and Zhu by space-time properness, and improves their pinching constant in dimensions two, five and six. For closed flows, we extend Lynch's planarity estimate to a larger range of exponents.Mean curvature flowAncient solutionsCurvature pinching https://arxiv.org/abs/2610.01713arxiv:2610.01713Thu, 01 Oct 2026 00:00:00 GMTDebora Impera, Michele Rimoldi, Francesco Ruatta — 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-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.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsCurvature pinchingStability & uniquenessMinimal & CMC surfaces https://arxiv.org/abs/2602.19681arxiv:2602.19681Mon, 23 Feb 2026 00:00:00 GMTDebora Impera, Michele Rimoldi, Francesco Ruatta — We classify properly immersed self-shrinkers of the mean curvature flow in arbitrary codimension under a quadratic pinching condition of Andrews-Baker type on the second fundamental form that is preserved along the flow. Under this assumption, such self-shrinkers reduce effectively to codimension one and are therefore generalized self-shrinking cylinders. In contrast to previous works, our approach is purely elliptic: it relies on parabolicity in a weighted setting and is tailored specifically to self-shrinkers, rather than to general ancient solutions of the flow. This allows us to avoid assuming any uniform pinching condition, to treat in any dimension the sharp Andrews-Baker pinching constant 4/3n and hence to sharpen, in the self-shrinker setting, the pinching constants appearing in recent classification results for ancient solutions.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsCurvature pinching