Complete classification of compact four-manifolds with positive isotropic curvature
Let \((M,<,>)\) be an \(n\)-dimensional Riemannian manifold. The curvature operator of \(M\) at a point \(p\) is the self-adjoint linear endomorphism of \(\Lambda ^2 T_p M\) defined by NEWLINE\[NEWLINE<\mathcal{R} (X \wedge Y),U\wedge V>=<R(X,Y)U,V>,NEWLINE\]NEWLINE where \(R\) is the curvature tensor. The Riemannian metric \(<,>\) extends to a Hermitian inner product \(<<,>>\) on \(T_p M \otimes \mathbb C\), and \(\mathcal{R}\) to a complex linear map on \(\Lambda ^2 T_p M \otimes \mathbb C\), denoted again by \(\mathcal{R}\). For any two-plane \(\sigma \subset T_p M \otimes \mathbb C\), the complex sectional curvature is defined by NEWLINE\[NEWLINEK_{\mathbb C} (\sigma)=<<\mathcal{R}(Z\wedge W),Z\wedge W>>,NEWLINE\]NEWLINE where \(\{Z,W\}\) is a basis of \(\sigma\), unitary with respect to \(<<,>>\). If \(K_{\mathbb C} (\sigma) >0\) for any totally isotropic plane at any point, then \(M\) is said to have positive isotropic curvature.NEWLINENEWLINEPositive isotropic curvature was introduced by \textit{M. J. Micallef} and \textit{J. D. Moore} [Ann. Math. (2) 127, No. 1, 199--227 (1988; Zbl 0661.53027)], who proved that a compact simply connected \(n(\geq 4)-\)dimen\-sional manifold with positive isotropic curvature is homeomorphic to a sphere. Schoen conjectured that an \(n(\geq 4)\)-dimensional compact manifold with positive isotropic curvature has a finite cover diffeomorphic to either \(\mathbb S^n\), \(\mathbb S^{n-1}\times \mathbb S^1\), or to a connected sum of them.NEWLINENEWLINEIn the present paper, this conjecture is proved in dimension four. More precisely, the following complete classification result is proved: a compact \(4\)-dimensional manifold admits a metric with positive isotropic curvature if and only if it is diffeomorphic to either \(\mathbb S^4\), \(\mathbb R \mathbb P ^4\), \(\mathbb S^3 \times \mathbb R /G\), or a connected sum of them, where \(G\) is a cocompact fixed-point free discrete subgroup of the isometry group of the standard metric on \(\mathbb S ^3 \times \mathbb R\).NEWLINENEWLINEAs a consequence of this classification, the authors also conclude that a \(4\)-dimensional compact manifold admits a metric of positive isotropic curvature if and only if it admits a conformally flat metric of positive scalar curvature.NEWLINENEWLINEThe proof of the main result uses first the Hamilton-Perelman theory on the Ricci flow with surgery: a compact \(4\)-manifold with positive isotropic curvature is taken as initial data and evolved by the Ricci flow. The manifold is then proved to be diffeomorphic to an orbifold connected sum of spherical orbifolds. In the second part of the proof, the authors show how to recover the topology of the manifold from the orbifold connected sum.
scientific article; zbMATH DE number 6081388