Paper · May 29, 2023
Classification of compact manifolds with positive isotropic curvature
As on arXiv
Abstract
We show the following result: Let be a compact manifold of dimension with positive isotropic curvature. Then is diffeomorphic to a spherical space form, or a quotient manifold of by a cocompact discrete subgroup of the isometry group of the round cylinder , or a connected sum of a finite number of such manifolds. This extends previous works of Brendle and Chen-Tang-Zhu, and improves a work of Huang. The proof uses Ricci flow with surgery on compact orbifolds, with the help of the ambient isotopy uniqueness of closed tubular neighborhoods of an isolated singular point in an orbifold.
Comments: 25 pages, restated Lemma 2.1 and Proposition 2.2 and rewrote their proofs
Its neighbourhood
Connected papers
The 8 papers on this site closest to this one, judged by the topics they share, the words and phrases in their titles and abstracts, and their authors. They are not its citations, but a map of the work around it. Papers here go back to 2021, plus the canon.
-
Open manifolds with uniformly positive isotropic curvature
Topics Surgery, Sphere theorems · Same author Hong Huang · Shared terms “positive isotropic”, “isotropic curvature”, “discrete subgroup”
-
Ricci flow on manifolds with positive isotropic curvature in all dimensions
Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”
-
Manifolds with 1/4-pinched Curvature are Space Forms
Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”, “space forms”
-
On Shrinking Ricci solitons with positive isotropic curvature in higher dimensions
Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”, “quotient”
-
Small Normal Curvature and Three-Manifold Topology
Topics Surgery, Sphere theorems · Shared terms “space forms”, “diffeomorphic”, “Brendle”
-
Pinching cones for positive isotropic curvature in dimensions seven and eight
Topics Sphere theorems · Shared terms “positive isotropic”, “isotropic curvature”, “compact manifolds”
-
Rigidity of area non-increasing maps
Topics Sphere theorems, Stability & uniqueness · Shared terms “isotropic curvature”, “isometry”, “Brendle”
-
Uniqueness and stability of singular Ricci flows in higher dimensions
Topics Sphere theorems, Stability & uniqueness · Shared terms “positive isotropic”, “isotropic curvature”, “Chen”
Read around it
Background reading
The guides for the topics of this paper: what each idea is, why it matters, and the papers to read first.
In the guidebook
- 11A.11Hamilton’s 1982 Theorem
Positive Ricci curvature in dimension 3 flows to a round metric.
- 11B.7Hamilton’s Program in 2002
What was proved, what was missing, and where Perelman came in.
- 12B.5The Canonical Neighbourhood Theorem
Every high-curvature point looks like a neck or a cap.