Stephen Lynch — Neckpinch

The planarity estimate in dimensions five and six

Wed, 30 Sep 2026 00:00:00 GMT

Stephen Lynch — Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews–Baker pinching range in dimensions at least 7, while in dimensions 5 and 6 a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions 5 and 6 throughout the full Andrews–Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.

Rotational symmetry of ancient solutions to fully nonlinear curvature flows

Thu, 12 Oct 2023 00:00:00 GMT

A. Cogo, S. Lynch, O. Vičánek Martínez — We address the classification of ancient solutions to fully nonlinear curvature flows for hypersurfaces. Under natural conditions on the speed of motion we classify ancient solutions which are convex, noncollapsing, uniformly two-convex and noncompact. There are exactly two possibilities – every such solution is either a self-similarly shrinking cylinder, or else is a rotationally symmetric translating soliton. For a large class of flows this yields a complete classification of the blow-up limits that can arise at a singularity of a solution which is compact, embedded and two-convex.

Hamilton's Theorem (on the compactness of pinched hypersurfaces) via mean curvature flow

Tue, 22 Nov 2022 00:00:00 GMT

Theodora Bourni, Mat Langford, Stephen Lynch — We make rigorous an old idea of using mean curvature flow to prove a theorem of Richard Hamilton on the compactness of proper hypersurfaces with pinched, bounded curvature.

Ancient solutions of Ricci flow with Type I curvature growth

Fri, 11 Nov 2022 00:00:00 GMT

Stephen Lynch, Andoni Royo Abrego — Ancient solutions of the Ricci flow arise naturally as models for singularity formation. There has been significant progress towards the classification of such solutions under natural geometric assumptions. Nonnegatively curved solutions in dimensions 2 and 3, and uniformly PIC solutions in higher dimensions are now well understood. We consider ancient solutions of arbitrary dimension which are complete and have Type I curvature growth. We show that a κ-noncollapsed Type I ancient solution which is noncompact and has nonnegative sectional curvature necessarily splits at least one Euclidean factor. It follows that a κ-noncollapsed Type I ancient solution which is weakly PIC2 is a locally symmetric space.

Quadratically pinched submanifolds of the sphere via mean curvature flow with surgery

Wed, 08 Sep 2021 00:00:00 GMT

Mat Langford, Stephen Lynch, Huy The Nguyen — We study mean curvature flow of n-dimensional submanifolds of S_K^(n+ℓ), the round (n+ℓ)-sphere of sectional curvature K>0, under the quadratic curvature pinching condition |A|² < 1/(n-2)|H|² + 4K when n≥ 8, |A|² < 4/3n|H|²+n/2K when n=7, and |A|²<(3(n+1))/(2n(n+2))|H|²+(2n(n-1))/(3(n+1))K when n=5 or 6. This condition is related to a theorem of Li and Li [Arch. Math., 58:582–594, 1992] which states that the only n-dimensional minimal submanifolds of S_K^(n+ℓ) satisfying |A|²<2n/3K are the totally geodesic n-spheres. We prove the existence of a suitable mean curvature flow with surgeries starting from initial data satisfying the pinching condition. As a result, we conclude that any smoothly, properly immersed submanifold of S_Kⁿ⁺¹ satisfying the pinching condition is diffeomorphic either to the sphere Sⁿ or to the connected sum of a finite number of handles S¹× Sⁿ⁻¹. The results are sharp when n≥ 8 due to hypersurface counterexamples.