Ancient solutions to the Ricci flow with isotropic curvature conditions. (English) Zbl 1527.53087
A Riemannian manifold is said to have positive isotropic curvature (PIC) if the complex sectional curvatures are positive for all isotropic two-planes. This condition was introduced by M. J. Micallef and J. D. Moore [Ann. Math. (2) 127, No. 1, 199–227 (1988; Zbl 0661.53027)], who proved that a compact simply connected Riemannian manifold with PIC is homeomorphic to a sphere. The fact that PIC is preserved by the Ricci flow, proved by R. S. Hamilton [Commun. Anal. Geom. 5, No. 1, 1–92 (1997; Zbl 0892.53018)] in dimension four and independently by S. Brendle and R. Schoen [J. Am. Math. Soc. 22, No. 1, 287–307 (2009; Zbl 1251.53021)] and H. T. Nguyen [Int. Math. Res. Not. 2010, No. 3, 536–558 (2010; Zbl 1190.53068)] in higher dimensions, played a central role in the proof of the celebrated quarter-pinched differentiable sphere theorem. An important question in the study of PIC is to classify compact Riemannian manifolds with PIC using Ricci flow with surgery.
This paper studies ancient solutions to the Ricci flow with PIC. The main result states that every \(\kappa\)-noncollapsed, complete noncompact, ancient Ricci flow with uniformly PIC for \(n=4\) or \(n\geq 12\), must have weakly PIC2 and bounded curvature. Combined with a result of S. Brendle and K. Naff [Geom. Topol. 27, No. 1, 153–226 (2023; Zbl 07688327)], this gives a full classification of such ancient Ricci flows: either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton.
The proof has two parts. First, ancient Ricci flows satisfying the above-mentioned conditions have weakly PIC2. Using Hamilton’s pinching estimate, the authors prove that the curvature operator of nonnegative in dimension four. In dimensions \(n\geq 12\), the proof uses the continuous family of invariant curvature cones constructed by S. Brendle [Ann. Math. (2) 190, No. 2, 465–559 (2019; Zbl 1423.53080)]. The second part is to prove such ancient Ricci flows have bounded curvature. This relies on a version of the canonical neighborhood theorem for Ricci flows with uniformly PIC and weakly PIC2.
In addition, this paper classifies complex two-dimensional, \(\kappa\)-noncollapsed, complete noncompact ancient Kähler-Ricci flows with weakly PIC1.
This paper studies ancient solutions to the Ricci flow with PIC. The main result states that every \(\kappa\)-noncollapsed, complete noncompact, ancient Ricci flow with uniformly PIC for \(n=4\) or \(n\geq 12\), must have weakly PIC2 and bounded curvature. Combined with a result of S. Brendle and K. Naff [Geom. Topol. 27, No. 1, 153–226 (2023; Zbl 07688327)], this gives a full classification of such ancient Ricci flows: either a family of shrinking cylinders (or a quotient thereof) or the Bryant soliton.
The proof has two parts. First, ancient Ricci flows satisfying the above-mentioned conditions have weakly PIC2. Using Hamilton’s pinching estimate, the authors prove that the curvature operator of nonnegative in dimension four. In dimensions \(n\geq 12\), the proof uses the continuous family of invariant curvature cones constructed by S. Brendle [Ann. Math. (2) 190, No. 2, 465–559 (2019; Zbl 1423.53080)]. The second part is to prove such ancient Ricci flows have bounded curvature. This relies on a version of the canonical neighborhood theorem for Ricci flows with uniformly PIC and weakly PIC2.
In addition, this paper classifies complex two-dimensional, \(\kappa\)-noncollapsed, complete noncompact ancient Kähler-Ricci flows with weakly PIC1.
Reviewer: Xiaolong Li (Wichita)
MSC:
| 53E20 | Ricci flows |
| 53E30 | Flows related to complex manifolds (e.g., Kähler-Ricci flows, Chern-Ricci flows) |
Citations:
Zbl 0661.53027; Zbl 0892.53018; Zbl 1251.53021; Zbl 1190.53068; Zbl 07688327; Zbl 1423.53080References:
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