Manifolds with positive curvature operators are space forms

By Christoph E. Boehm, Burkhard Wilking. 2006

395 citations recorded by the source corpus.

The Ricci flow was introduced by Hamilton in 1982 [H1] in order to prove that a compact three-manifold admitting a Riemannian metric of positive Ricci curvature is a spherical space form. In dimension four Hamilton showed that compact four-manifolds with positive curvature operators are spherical space forms as well [H2]. More generally, the same conclusion holds for compact four-manifolds with 2-positive curvature operators [Che]. Recall that a curvature operator is called 2-positive, if the sum of its two smallest eigenvalues is positive. In arbitrary dimensions Huisken [Hu] described an explicit open cone in the space of curvature operators such that the normalized Ricci flow evolves metrics whose curvature operators are contained in that cone into metrics of constant positive sectional curvature. Hamilton conjectured that in all dimensions compact Riemannian manifolds with positive curvature operators must be space forms. In this paper we confirm this conjecture. More generally, we show the following

Subfields: Applied Mathematics, Geometry and Topology, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/72013cce6689f1e82d6e410dcadc4a81f3194f79

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

In Geometry and Topology

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

Manifolds with 1/4-pinched curvature are space forms (2007, methodology): “1 of Böhm-Wilking [3] can now be used to provide a pinching set as in Proposition 16.”
Ricci solitons: the equation point of view (2008, background): “Recently B¨ohm and Wilking [ 2 ] showed the following Hamilton’s conjecture.”
Recent Progress on Ricci Solitons (2009, methodology): “On the other hand, we know that compact shrinking solitons with positive curvature operator are isometric to finite quotients of round spheres, thanks to the works of Hamilton [62, 63] (for n = 3, 4) and Böhm-Wilking [4] (for n ≥ 5).”
Immortal homogeneous Ricci flows (2017, result): “itive Ricci curvature can be deformed to the round metric via volume normalized Ricci ﬂow [Ham82]. Similar convergence results were obtained afterwards in higher dimensions, see for instance [Ham86], [BW08] and [BS09]. About 20 years later, Perelman was able to describe new geometric quantities, whi”
Frankel conjecture and Sasaki geometry (2012, methodology): “Finally, as already pointed out in [4] a pinching towards constant curvature proof of the Frankel conjecture using Ricci flow seems not plausible, as there are examples of two dimensional Ricci soliton orbifolds with positive curvature.”
Local mollification of Riemannian metrics using Ricci flow, and Ricci limit spaces (2017, methodology): “duction in 1982 by Hamilton, Ricci ﬂow has become central to our eﬀorts to understand the topology of manifolds that are either low dimensional, or that satisfy a curvature condition. See for example [14, 22, 3, 4]. Loosely speaking, the strategy is to allow Ricci ﬂow to evolve such a manifold into ”
The Existence of Type II Singularities for the Ricci Flow on $S^{n+1}$ (2007, background): “By Theorem 1 in [7], we see that the compact gradient shrinking soliton is getting round”
Ricci flow with surgery on manifolds with positive isotropic curvature (2017, background): “It is shown in [1] that l−1 a,b(Q(la,b(S)) = Q(S) +Da,b(S), where Da,b(S) is defined by Da,b(S) = (2b+ (n− 2)b(2) − 2a) o Ric(S) ? o Ric(S) + 2aRic(S) ? Ric(S) + 2b(2) o Ric(S)(2) ? id”
An exotic sphere with positive sectional curvature (2008, methodology): “most complete classiﬁcation of manifolds with nonnegative curvature operator with important improvements by Mok-Siu-Yau and Micallef-Moore and most recently by B¨ohm-Wilking (see [Pet, Chapter 8] and [BohmWilk]). The upshot of this classiﬁcation is that any homotopy sphere with nonnegative curvature”
On Locally Conformally Flat Gradient Shrinking Ricci Solitons (2008, background): “f n >3, there exist shrinking Ricci solitons which are not Einstein (See H.-D. Cao [Cao96], N. Koiso [Koi90], M.Feldman, T. Ilmanen and D. Knopf [FIK03]). The recent work of C. Böhm and B. Wilking [BW06] implies that gradient Ricci solitons with positive curvature operator must be of constant cur”
Isotropic Curvature and the Ricci Flow (2010, background): “of Böhm and Wilking [2], who showed that manifolds with 2-positive curvature are diffeomorphic to space forms.”
Rigidity of asymptotically conical shrinking gradient Ricci solitons (2013, result): “The results of Böhm-Wilking [2], for example, imply that two-positivity of the curvature operator is sufficient, and the results of Brendle-Schoen [8] and Brendle [5] demonstrate that this can be relaxed further still, e.”
Geometric, topological and differentiable rigidity of submanifolds in space forms (2011, background): “[3] C. Böhm and B. Wilking, Manifolds with positive curvature operator are space forms, Ann. of Math., 167(2008), 1079-1097.”