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

Mat Langford; Huy The Nguyen

Paper · Jun 14, 2020

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

Mat Langford, Huy The Nguyen

math.DGWider flowsarXiv:2006.08049

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Abstract

We study mean curvature flow in $S_{K}^{n + 1}$ , the round sphere of sectional curvature $K > 0$ , under the quadratic curvature pinching condition $∣ A ∣^{2} < \frac{1}{n - 2} H^{2} + 4 K$ when $n \geq 4$ and $∣ A ∣^{2} < \frac{3}{5} H^{2} + \frac{8}{3} K$ when $n = 3$ . This condition is related to a famous theorem of Simons, which states that the only minimal hypersurfaces satisfying $∣ A ∣^{2} < n K$ are the totally geodesic hyperspheres. It is related to but distinct from two-convexity. Notably, in contrast to two-convexity, it allows the mean curvature to change sign. We show that the pinching condition is preserved by mean curvature flow, and obtain a cylindrical estimate and corresponding pointwise derivative estimates for the curvature. As a result, we find that the flow becomes either uniformly convex or quantitatively cylindrical in regions of high curvature. This allows us to apply the surgery apparatus developed by Huisken and Sinestrari. We conclude that any smoothly, properly, isometrically immersed hypersurface $M$ of $S_{K}^{n + 1}$ satisfying the pinching condition is diffeomorphic to $S^{n}$ or the connected sum of a finite number of copies of $S^{1} \times S^{n - 1}$ . If $M$ is embedded, then it bounds a 1-handlebody. The results are sharp when $n \geq 4$ .

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Submitted: Jun 14, 2020
Subjects: math.DG

Mean curvature flow Surgery Curvature pinching Minimal & CMC surfaces

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The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.

Mean curvature flow Wider flows A hypersurface moves in the direction of its mean curvature vector, which decreases area as fast as possible in the L² sense. 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.

[1] 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).

[2] 11B.7
Hamilton’s Program in 2002Book 11B · Solitons, Compactness and Singularities
What was proved, what was missing, and where Perelman came in.

[3] 12B.5
The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery
Every high-curvature point looks like a neck or a cap.

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

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