https://neckpinch.com/authors/yi-lai/New arXiv papers by Yi Lai, from Neckpinch Tue, 03 Nov 2026 07:15:30 GMT https://arxiv.org/abs/2512.25050arxiv:2512.25050Wed, 31 Dec 2025 00:00:00 GMTRichard H. Bamler, Yi Lai — We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the PDE–ODI principle, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the Łojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.Mean curvature flowRicci solitonsAncient solutionsSelf-shrinkers & translatorsSingularitiesStability & uniqueness https://arxiv.org/abs/2512.24524arxiv:2512.24524Wed, 31 Dec 2025 00:00:00 GMTRichard H. Bamler, Yi Lai — We resolve the Mean Convex Neighborhood Conjecture for mean curvature flows in all dimensions and for all types of cylindrical singularities. Specifically, we show that if the tangent flow at a singular point is a multiplicity-one cylinder, then in a neighborhood of that point the flow is mean-convex, its time-slices arise as level sets of a continuous function, and all nearby tangent flows are cylindrical. Moreover, we establish a canonical neighborhood theorem near such points, which characterizes the flow via local models. We also obtain a more uniform version of the Mean Convex Neighborhood Conjecture, which only requires closeness to a cylinder at some initial time and yields a quantitative version of this structural description. Our proof relies on a complete classification of ancient, asymptotically cylindrical flows. We prove that any such flow is non-collapsed, convex, rotationally symmetric, and belongs to one of three canonical families: ancient ovals, the bowl soliton, or the flying wing translating solitons. Central to our method is a refined asymptotic analysis and a novel leading mode condition, together with a new "induction over thresholds" argument. In addition, our approach provides a full parameterization of the space of asymptotically cylindrical flows and gives a new proof of the existence of flying wing solitons. Our method is independent of prior work and, together with our prequel paper, this work is largely self-contained.Mean curvature flowAncient solutionsSelf-shrinkers & translatorsSingularities(Non)collapsingRotational symmetry & warped products https://arxiv.org/abs/2510.23005arxiv:2510.23005Mon, 27 Oct 2025 00:00:00 GMTPak-Yeung Chan, Yi Lai, Man-Chun Lee — For any n≥ 4, we construct an (n-2)-parameter family of steady gradient Ricci solitons with non-negative curvature operator and prescribed by the eigenvalues of Ricci tensor at a critical point of the soliton potential. Among them lies an (n-3)-parameter subfamily of non-collapsed solitons. These solitons generalized the flying wings constructed by the second named author and produced new examples of steady gradient Ricci solitons with non-negative curvature operator for n≥ 4. Our approach is based on constructing continuous families of Ricci flows smoothing emanating from continuous families of spherical polyhedra which still preserves symmetry. This is built upon a new stability result of Ricci flows with scaling invariant estimates. As another application of the method, we prove the stability of asymptotically conical expanding solitons constructed by Deruelle under L^∞ perturbation of links. In particular, the C⁰-convergence of smooth links implies the smooth convergence of the expanding solitons.Ricci flowRicci solitons(Non)collapsingStability & uniquenessBoundaries & non-compactEigenvalues & spectrum https://arxiv.org/abs/2504.02804arxiv:2504.02804Thu, 03 Apr 2025 00:00:00 GMTKyeongsu Choi, Yi Lai — For an ancient Ricci flow asymptotic to a compact integrable shrinker, or a Ricci flow developing a finite-time singularity modelled on the shrinker, we establish the long-time existence of a harmonic map heat flow between the Ricci flow and the shrinker for all times. This provides a global parabolic gauge for the Ricci flow and implies the uniqueness of the tangent flow without modulo any diffeomorphisms. We present two main applications: First, we construct and classify all ancient Ricci flows asymptotic to any compact integrable shrinker, showing that they converge exponentially. Second, we obtain the optimal convergence rate at singularities modelled on the shrinker, characterized by the first negative eigenvalue of the stability operator for the entropy. In particular, we show that any Ricci flow developing a round Sⁿ singularity converges at least at the rate (−t)^((n+1)/(n-1)).Ricci flowRicci solitonsAncient solutionsSingularitiesStability & uniquenessLong-time existence & convergenceHarmonic maps & DeTurck trickEigenvalues & spectrum https://arxiv.org/abs/2502.09727arxiv:2502.09727Thu, 13 Feb 2025 00:00:00 GMTOtis Chodosh, Yi Lai, Kai Xu — We show that a complete contractible 3-manifold with positive scalar curvature and bounded geometry must be ℝ³. We also show that an open handlebody of genus larger than 1 does not admit complete metrics with positive scalar curvature and bounded geometry. Our results rely on the maximal weak solution to inverse mean curvature flow due to the third-named author.Mean curvature flowOther curvature flowsInverse mean curvature flow3-manifoldsScalar curvatureWeak & singular flows https://arxiv.org/abs/2403.04089arxiv:2403.04089Wed, 06 Mar 2024 00:00:00 GMTPak-Yeung Chan, Ronan J. Conlon, Yi Lai — In 1996, H.-D. Cao constructed a U(n)-invariant steady gradient Kähler-Ricci soliton on ℂⁿ and asked whether every steady gradient Kähler-Ricci soliton of positive curvature on ℂⁿ is necessarily U(n)-invariant (and hence unique up to scaling). Recently, Apostolov-Cifarelli answered this question in the negative for n=2. Here, we construct a family of U(1)× U(n-1)-invariant, but not U(n)-invariant, complete steady gradient Kähler-Ricci solitons with strictly positive curvature operator on real (1, 1)-forms (in particular, with strictly positive sectional curvature) on ℂⁿ for n≥3, thereby answering Cao's question in the negative for n≥3. This family of steady Ricci solitons interpolates between Cao's U(n)-invariant steady Kähler-Ricci soliton and the product of the cigar soliton and Cao's U(n-1)-invariant steady Kähler-Ricci soliton. This provides the Kähler analog of the Riemannian flying wings construction of Lai. In the process of the proof, we also demonstrate that the almost diameter rigidity of ℙⁿ endowed with the Fubini-Study metric does not hold even if the curvature operator is bounded below by 2 on real (1, 1)-forms.Ricci solitonsKähler–Ricci solitonsAncient solutionsSphere theoremsKähler geometryStability & uniqueness https://arxiv.org/abs/2207.02714arxiv:2207.02714Wed, 06 Jul 2022 00:00:00 GMTYi Lai — For every θ∈(0,π), we construct a 3D steady gradient Ricci soliton whose asymptotic cone is a sector with angle θ, which is a called 3D flying wing.Ricci solitons3-manifolds https://arxiv.org/abs/2205.01146arxiv:2205.01146Mon, 02 May 2022 00:00:00 GMTYi Lai — For any 3D steady gradient Ricci soliton with positive curvature, we prove that it must be isometric to the Bryant soliton if it is asymptotic to a ray. Otherwise, it is asymptotic to a sector and hence a flying wing. We show that all 3D flying wings are O(2)-symmetric. Therefore, all 3D steady gradient Ricci solitons are O(2)-symmetric.Ricci solitonsAncient solutions3-manifolds