Paper · Sep 8, 2021

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

Mat Langford, Stephen Lynch, Huy The Nguyen

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Abstract

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 $∣ A ∣^{2} < \frac{1}{n - 2} ∣ H ∣^{2} + 4 K$ when $n \geq 8$ , $∣ A ∣^{2} < \frac{4}{3 n} ∣ H ∣^{2} + \frac{n}{2} K$ when $n = 7$ , and $∣ A ∣^{2} < \frac{3 ( n + 1 )}{2 n ( n + 2 )} ∣ H ∣^{2} + \frac{2 n ( n - 1 )}{3 ( n + 1 )} K$ 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 $∣ A ∣^{2} < \frac{2 n}{3} K$ 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 $S_{K}^{n + 1}$ satisfying the pinching condition is diffeomorphic either to the sphere $S^{n}$ or to the connected sum of a finite number of handles $S^{1} \times S^{n - 1}$ . The results are sharp when $n \geq 8$ due to hypersurface counterexamples.

Comments: arXiv admin note: text overlap with arXiv:2006.08049

Mean curvature flow Surgery Curvature pinching Minimal & CMC surfaces

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Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible. Start with: Flow by mean curvature of convex surfaces into spheres (Huisken, 1984) Surgery Stop the flow just before a neck pinches, cut along the thin neck, glue standard caps onto the two ends, and restart. Start with: Four-manifolds with positive isotropic curvature (Hamilton, 1997) Curvature pinching Curvature conditions that Ricci flow preserves and improves. Hamilton–Ivey pinching makes positive curvature dominate in 3D blow-ups, and invariant… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Minimal & CMC surfaces Minimal surfaces are critical points of area, with zero mean curvature; constant mean curvature (CMC) surfaces are critical for area at fixed volume. Start with: On the structure of manifolds with positive scalar curvature (Schoen et al., 1979)

In the guidebook

12B.10
Surgery on a δ-NeckBook 12B · κ-Solutions and Surgery
The gluing construction in detail: the profile function, preservation of the Hamilton–Ivey pinching, closeness to the standard solution, and the definition of Ricci flow with surgery as a space-time (Morgan–Tian chapters 13–15).
11B.7
Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities
What was proved, what was missing, and where Perelman came in.
12B.5
The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery
Every high-curvature point looks like a neck or a cap.

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

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