https://neckpinch.com/authors/michele-rimoldi/New arXiv papers by Michele Rimoldi, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT 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 https://arxiv.org/abs/2310.12722arxiv:2310.12722Thu, 19 Oct 2023 00:00:00 GMTDebora Impera, Michele Rimoldi — We prove an abstract structure theorem for weighted manifolds supporting a weighted f-Poincaré inequality and whose ends satisfy a suitable non-integrability condition. We then study how our arguments can be used to obtain full topological control on two important classes of hypersurfaces of the Euclidean space, namely translators and self-expanders for the mean curvature flow, under either stability or curvature asumptions. As an important intermediate step in order to get our results we get the validity of a Poincaré inequality with respect to the natural weighted measure on any translator and we prove that any end of a translator must have infinite weighted volume. Similar tools can be obtained for properly immersed self-expanders permitting to get topological rigidity under curvature assumptions.Mean curvature flowSelf-shrinkers & translatorsStability & uniqueness