Paper · Nov 27, 2023

Open manifolds with uniformly positive isotropic curvature

As on arXiv

Abstract

We prove the following result: Let $(M, g_{0})$ be a complete noncompact manifold of dimension $n \geq 12$ with isotropic curvature bounded below by a positive constant, with scalar curvature bounded above, and with injectivity radius bounded below. Then there is a finite collection $X$ of spherical $n$ -manifolds and manifolds of the form $S^{n - 1} \times R / G$ , where $G$ is a discrete subgroup of the isometry group of the round cylinder $S^{n - 1} \times R$ , such that $M$ is diffeomorphic to a (possible infinite) connected sum of members of $X$ . This extends a recent work of Huang. The proof uses Ricci flow with surgery on open orbifolds with isolated singularities.

Comments: arXiv admin note: text overlap with arXiv:1909.12265

Ricci flow Singularities Surgery Sphere theorems Scalar curvature Boundaries & non-compact

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Ricci flow Hamilton's Ricci flow evolves a Riemannian metric by minus twice its Ricci curvature. It is a nonlinear heat equation for geometry: it spreads… Start with: Three-manifolds with positive Ricci curvature (Hamilton, 1982) Singularities On a closed manifold the flow usually cannot run forever: curvature blows up in finite time, often because a neck pinches off. Start with: The formations of singularities in the Ricci Flow (Hamilton, 1993) 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) Sphere theorems Curvature conditions that force a manifold to be a sphere. Ricci flow settled the 1/4-pinched differentiable sphere theorem, which had been open for… Start with: Minimal Two-Spheres and the Topology of Manifolds with Positive Curvature on Totally Isotropic Two-Planes (Micallef et al., 1988)

In the guidebook

12B.5
The Canonical Neighbourhood TheoremBook 12B · κ-Solutions and Surgery
Every high-curvature point looks like a neck or a cap.
12B.7
The Singular Time: Horns and Strong NecksBook 12B · κ-Solutions and Surgery
The limit metric at a singular time, the set Ω where curvature stays bounded, ε-horns and their ends, and strong δ-necks deep in a horn, following Perelman II §3 and Morgan–Tian chapter 11.
12B.12
Canonical Neighbourhoods After SurgeryBook 12B · κ-Solutions and Surgery
The strong canonical neighbourhood assumption justified across surgeries: the blow-up argument, the three cases for a limit that meets a surgery, and the choice of δ (Perelman II §5; Morgan–Tian chapter 17).

Open manifolds with uniformly positive isotropic curvature

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