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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.
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-ma
In this note we prove that a four-dimensional compact oriented half-confor-mally flat Riemannian manifold $M^4$ is topologically $mathbb{S}^{4}$ or $mathbb{C}mathbb{P}^{2},$ provided that the sectional curvatures all lie in the interval $[frac{3sqrt{
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
The critical point equation arises as a critical point of the total scalar curvature functional defined on the space of constant scalar curvature metrics of a unit volume on a compact manifold. In this equation, there exists a function $f$ on the man
We prove that a complete noncompact K{a}hler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {bf C}$^{n}$ and we show that the manifold is topologically {bf R}$^{2n}