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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 Weyl tensor with the non-negativity of the complete divergence of the Bach tensor implies the harmonicity of the metric, and we present 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. Second, we prove the Besse conjecture under these conditions, which are weaker than harmonicity or Bach flatness of the metric. Moreover, we show a rigidity result for vacuum static spaces and find a sufficient condition for the metric to be Bach-flat.

In this paper, we prove rigidity results on gradient shrinking Ricci solitons with weakly harmonic Weyl curvature tensors. Let $(M^n, g)$ be a compact gradient shrinking Ricci soliton satisfying ${\rm Ric}_g + Ddf = \rho g$ with $\rho >0$ constant. We show that if $(M,g)$ satisfies $\delta \mathcal W (\cdot, \cdot, \nabla f) = 0$, then $(M, g)$ is Einstein. Here $\mathcal W$ denotes the Weyl curvature tensor. In the case of noncompact, if $M$ is complete and satisfies the same condition, then $M$ is rigid in the sense that $M$ is given by a quotient of product of an Einstein manifold with Euclidean space. These are generalizations of the previous known results in \cite{l-r}, \cite{m-s} and \cite{p-w3}., Comment: 15pages

In this paper, we study the structure of generalized Bach-flat vacuum static spaces. Generalized Bach-flat metrics are considered as extensions of both Einstein and Bach-flat metrics. First, we prove that a compact Riemannian n -manifold with n ≥ 4 which is a generalized Bach-flat vacuum static space is Einstein. A generalized Bach-flat vacuum static space with the potential function f having compact level sets is either Ricci-flat or a warped product with zero scalar curvature when n ≥ 5 , and when n = 4 , it is Einstein if f has its minimum. Secondly, we consider critical metrics for another quadratic curvature functional involving the Ricci tensor, and prove similar results. Lastly, by applying the technique developed above, we prove Besse conjecture when the manifold is generalized 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 curvature, 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 $n \geq 5$, and a similar condition for $n = 4$.

Let N be a complete Riemannian manifold with nonnegativeRicci curvature and let M be a complete noncompact oriented stableminimal hypersurface in N. We prove that if M has at least two endsandR M |A| 2 dv = ∞, then M admits a nonconstant harmonic functionwith finite Dirichlet integral, where Ais the second fundamental form ofM. We also show that the space of L 2 harmonic 1-forms on such a stableminimal hypersurface is not trivial. Our result is a generalization of oneof main results in [12] because if N has nonnegative sectional curvature,then M admits no nonconstant harmonic functions with finite Dirichletintegral. And our result recovers a main theorem in [3] as a corollary. 1. IntroductionThe classical Bernstein theorem asserts that an entire minimal graph in R 3 must be planar. This theorem was subsequently generalized to higher dimen-sions by several authors, cf. [1], [5], [9], [17]. It is now known [2], [17] that anentire n-dimensional minimal graph in R n+1 must be given by a linear functionover R

In this paper, we prove that there are no multiple black holes in an n-dimensional static vacuum space-time having weakly harmonic curvature unless the Ricci curvature is trivial.

It has been realized that the proof of Theorem 5.1 in Section 5 is imcomplete. It was pointed out by Professor Jongsu Kim and Israel Evangelista. Here we give a correct proof of Theorem 5.1

On an $n$-dimensional complete manifold $M$, consider an $h$-almost gradient Ricci soliton, which is a generalization of a gradient Ricci soliton. We prove that if the manifold is Bach-flat and $dh/du>0$, then the manifold $M$ is either Einstein or rigid. In particular, such a manifold has harmonic Weyl curvature. Moreover, if the dimension of $M$ is four, the metric $g$ is conformally flat., 12 pages

In this paper, we deal with a critical point metric of the total scalar curvature on a compact manifold M. We prove that if the criti- cal point metric has parallel Ricci tensor, then the manifold is isometric to a standard sphere. Moreover, we show that if an n-dimensional Rie- mannian manifold is a warped product, or has harmonic curvature with non-parallel Ricci tensor, then it cannot be a critical point metric.

On a compact n -dimensional manifold M, it was shown that a critical point metric g of the total scalar curvature functional, restricted to the space of metrics with constant scalar curvature of volume 1, satisfies the critical point equation ([5], p. 3222). In 1987 Besse proposed a conjecture in his book [1], p. 128, that a solution of the critical point equation is Einstein (Conjecture A, hereafter). Since then, number of mathematicians have contributed for the proof of Conjecture A and obtained many geometric consequences as its partial proofs. However, none has given its complete proof yet. The purpose of the present paper is to prove Theorem 1, stating that a compact 3-dimensional manifold M is isometric to the round 3-sphere S3 if ker s′*g ≠ 0 and its second homology vanishes. Note that this theorem implies that M is Einstein and hence that Conjecture A holds on a 3-dimensional compact manifold under certain topological conditions (© 2010 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Let M n be a complete oriented non-compact minimally im- mersed submanifold in a complete Riemannian manifold N n+p of non- negative curvature. We prove that if M is super-stable, then there are no non-trivial L2 harmonic one forms on M. This is a generalization of the

For a compact $n$-dimensional manifold a critical point metric of the total scalar curvature functional satisfies the critical point equation (1) below, if the functional is restricted to the space of constant scalar curvature metrics of unit volume. The right-hand side in this equation is nothing but the adjoint operator of the linearization of the total scalar curvature acting on functions. The structure of the kernel space of the adjoint operator plays an important role in the geometry of the underlying manifold. ¶ In this paper, we study some geometric structure of a given manifold when the kernel space of the adjoint operator is nontrivial. As an application, we show that if there are two distinct solutions satisfying the critical point equation mentioned above, then the metric should be Einstein. This generalizes a main result in [6] to arbitrary dimension.

We discuss the harmonicity of horizontally conformal maps and their relations with the spectrum of the Laplacian. We prove that if φ : M → N is a horizontally conformal map such that the tension field is divergence free, then φ is harmonic. Furthermore, if N is noncompact, then φ must be constant. Also we show that the projection of a warped product manifold onto the first component is harmonic if and only if the warping function is constant. Finally, we describe a characterization for a horizontally conformal map with a constant dilation preserving an eigenfunction. We describe here some characterizations for the harmonicity of horizontally con- formal maps. In particular, we consider the projections of a warped product man- ifold onto each component manifold. Those are examples of horizontally confor- mal maps. We show that the projection of a warped product manifold onto the first component is a horizontally conformal map if and only if the warping function is constant. Finally, we consider the spectrum of the Laplacian and its relations with horizontally conformal maps. In (6), Gilkey and Park studied the spectrum of the Laplacian and Riemannian submersions. They proved that a Riemannian submersion φ : M → N commutes with the Laplacian if and only if φ ∗ preserves the eigenfunctions of the Laplacian. In (10), the author showed that, for horizontally conformal maps, a similar result hold. If a horizontally conformal map preserves an eigenfunction, then the dilation of the horizontally conformal map is given by the square root of the ratio of eigenvalues or a geometric identity must hold. Throughout, every manifold is connected and smooth, and a compact manifold is assumed to be compact without boundary otherwise stated.

On a compact n-dimensional manifold, it has been conjectured that a critical point metric of the total scalar curvature, restricted to the space of metrics with constant scalar curvature of unit volume, will be Einstein. This conjecture was proposed in 1984 by Besse, but has yet to be proved. In this paper, we prove that if the manifold with the critical point metric has harmonic curvature, then it is isometric to a standard sphere., 16 pages, 2 figures

Let φ: M → N be a harmonic morphism from a complete noncompact Riemannian manifold M with nonnegative Ricci curvature to a complete Riemannian manifold N with nonpositive scalar curvature. We show if the energy of φ is finite, then φ is constant. This can be compared with a similar result for harmonic maps when N has nonpositive sectional curvature due to Schoen and Yau.

We study the structure of manifolds with almost nonnegative Ricci curvature. We prove a compact Riemannian manifold with bounded curvature, diameter bounded from above, and Ricci curvature bounded from below by an almost nonnegative real number such that the first Betti number havingcodimension two is an infranilmanifold or a finite cover is a sphere bundle over a torus. Furthermore, if we assume the Ricci curvature is bounded and volume is bounded from below, then the manifold must be an infranilmanifold.