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Register a new userThe fundamental group of manifolds of positive isotropic curvature and surface groups
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In this paper we study the topology of compact manifolds of positive isotropic curvature (PIC). There are many examples of non-simply connected compact manifolds with positive isotropic curvature. We prove that the fundamental group of a compact Riemannian manifold with PIC, of dimension greater than or equal to 5, does not contain a subgroup isomorphic to the fundamental group of a compact Riemann surface. The proof uses stable minimal surface theory.
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Let M be a Riemannian n-manifold with n greater than or equal to 3. For k between 1 and n, we say M has k-positive Ricci curvature if at every point of M the sum of any k eigenvalues of the Ricci curvature is strictly positive. In particular, one positive Ricci curvature is equivalent to positive Ricci curvature and n-positive Ricci curvature is equivalent to positive scalar curvature. Let G be the fundamental group of the closed manifold M. We say that G is virtually free if G contains a free subgroup of finite index, or equivalently, if some finite cover of M has a fundamental group that is a free group. In this paper we will prove: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, such that (n-1)-eigenvalues of the Ricci curvature are strictly positive. Then the fundamental group of M is virtually free. As an immediate consequence we have: Let M be a closed Riemannian n-manifold, with n greater than or equal to 3, with 2-positive Ricci curvature. Then the fundamental group of M is virtually free.
In this paper, we show that a closed $n$-dimensional generalized ($lambda, n+m)$-Einstein manifold with positive isotropic curvature and constant scalar curvature must be isometric to either a sphere ${Bbb S}^n$, or a product ${Bbb S}^{1} times {Bbb S}^{n-1}$ of a circle with an $(n-1)$-sphere, up to finite cover and rescaling.
In this paper, we completely classify all compact 4-manifolds with positive isotropic curvature. We show that they are diffeomorphic to $mathbb{S}^4,$ or $mathbb{R}mathbb{P}^4$ or quotients of $mathbb{S}^3times mathbb{R}$ by a cocompact fixed point free subgroup of the isometry group of the standard metric of $mathbb{S}^3times mathbb{R}$, or a connected sum of them.
We survey the results on fundamental groups of open manifolds with nonnegative Ricci curvature. We also present some open questions on this topic.
In this paper we study the Ricci flow on compact four-manifolds with positive isotropic curvature and with no essential incompressible space form. Our purpose is two-fold. One is to give a complete proof of Hamiltons classification theorem on four-manifolds with positive isotropic curvature and with no essential incompressible space form; the other is to extend some recent results of Perelman on the three-dimensional Ricci flow to four-manifolds. During the the proof we have actually provided, up to slight modifications, all necessary details for the part from Section 1 to Section 5 of Perelmans second paper on the Ricci flow.
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