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 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 functi
on $f: M to {Bbb R}$ 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 $kappa =0$.
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 man
ifold 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 first prove the $f$-mean curvature comparison in a smooth metric measure space when the Bakry-Emery Ricci tensor is bounded from below and $|f|$ is bounded. Based on this, we define a Myers-type compactness theorem by generalizing t
he results of Cheeger, Gromov, and Taylor and of Wan for the Bakry-Emery Ricci tensor. Moreover, we improve a result from Soylu by using a weaker condition on a derivative $f(t)$.
For Riemannian manifolds with a smooth measure $(M, g, e^{-f}dv_{g})$, we prove a generalized Myers compactness theorem when Bakry--Emery Ricci tensor is bounded from below and $f$ is bounded.
In this paper, we study vacuum static spaces with the complete divergence of the Bach tensor and Weyl tensor. First, we prove that the vanishing of complete divergence of the Bach tensor and Weyl tensor implies the harmonicity of the metric, and we p
resent examples in which these conditions do not imply Bach flatness. As an application, we prove the non-existence of multiple black holes in vacuum static spaces with zero scalar curvature. On the other hand, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.
On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvatur
e, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture, proposed in 1987 by Besse, has not been resolved except when $M$ has harmonic curvature or the metric is Bach flat. In this paper, we prove some gap properties under divergence-free Bach tensor condition for $ngeq 5$, and a similar condition for $n=4$.
On a compact $n$-dimensional manifold $M$, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume, is Einstein. It has been conjectured that a critical metric of the total scalar cur
vature, restricted to the space of metrics with constant scalar curvature of unit volume, will also be Einstein. It was shown that this conjecture is true when $M$ together with a critical metric has harmonic curvature or the metric is Bach flat. In this paper, we tried to prove this conjecture with a divergence-free Bach tensor.
On a compact $n$-dimensional manifold, it is well known that a critical metric of the total scalar curvature, restricted to the space of metrics with unit volume is Einstein. It has been conjectured that a critical metric of the total scalar curvatur
e, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1987 by Besse, but has yet to be proved. In this paper, we prove the Besse conjecture with a weaker condition than harmonic curvature for $ngeq 3$.
