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In this paper, we give a complete classification of critical metrics of the volume functional on a compact manifold $M$ with boundary $partial M$ having positive isotropic curvature. We prove that for a pair $(f, kappa)$ of a nontrivial smooth function $f: M to {Bbb R}$ and a nonnegative real number $kappa$, if $(M, g)$ having positive isotropic curvature satisfies $$ Ddf - (Delta f)g - f{rm Ric} = kappa g, $$ then $(M, g)$ is isometric to a geodesic ball in ${Bbb S}^n$ when $kappa >0$, and either $M$ isometric to ${Bbb S}^n_+$, or the product $I times {Bbb S}^{n-1}$, up to finite cover when $kappa =0$.
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In this paper, we study vacuum static spaces with positive isotropic curvature. We prove that if $(M^n, g, f)$, $n ge 4$, is a compact vacuum static space with positive isotropic curvature, then up to finite cover, $M$ is isometric to a sphere ${Bbb S}^n$ or the product of a circle ${Bbb S}^1$ with an $(n-1)$-dimensional sphere ${Bbb S}^{n-1}$.
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 manifold that satisfies the following $$ (1+f){rm Ric} = Ddf + frac{nf +n-1}{n(n-1)}sg. $$ It has been conjectured that if $(g, f)$ is a solution of the critical point equation, then $g$ is Einstein and so $(M, g)$ is isometric to a standard sphere. In this paper, we show that this conjecture is true if the given Riemannian metric has positive isotropic curvature.
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.
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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