Search

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 investigate the geometry of Einstein-type equation on a Riemannian manifold, unifying various particular geometric structures recently studied in the literature, such as critical point equation and vacuum static equation. We show various rigidity results of Einstein-type manifolds under assumptions of several curvature conditions.

In this paper, we prove that a complete noncompact submanifold in a weighted manifold with nonpositive sectional curvature has at least linear weighted volume growth. Moreover we obtain several sufficient conditions for f -minimal hypersurfaces to have infinite weighted volume. By using an f -Laplacian comparison result, we obtain a lower bound of the first eigenvalue for the f -Laplace operator on submanifolds in a weighted manifold. We also obtain vanishing results for L f 2 harmonic 1-forms on complete noncompact f -minimal hypersurfaces in a weighted manifold. Finally we prove a topological structure theorem for complete noncompact L f -stable f -minimal hypersurfaces via a Liouville-type theorem for f -harmonic functions with finite f -energy.

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

Let M n be a complete oriented noncompact hypersurface in a complete Riemannian manifold N n+1 of nonnegative sectional curvature with $${2 \leq n \leq 5}$$ . We prove that if M satisfies a stability condition, then there are no non-trivial L 2 harmonic one-forms on M. This result is a generalization of a well-known fact in the case when M is a stable minimally immersed hypersurface. As a consequence, we show that if the mean curvature of M is constant, then either M must have only one end or M splits into a product of $${\mathbb{R}}$$ and a compact manifold with nonnegative sectional curvature. In case $${n \geq 5}$$ , we also show that the same result holds if the absolute value of the mean curvature is less than or equal to the ratio of the norm of the second fundamental form to the dimension of a hypersurface.

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)

In this paper, we introduce the notion of dilation and fuzzy Lipschitz of a map from a fuzzy metric space into a fuzzy metric space and we prove continuity properties for such maps. We also define the notion of the fuzzy Lipschitz distance between two fuzzy metric spaces and show that two compact fuzzy metric spaces whose Lipschitz distance is zero is fuzzy isometric to each other. On the other hand, we introduce the concept of minimal slope of a map between fuzzy metric spaces, which is defined by the ratio of two fuzzy metrics and derive some properties on it and relations with the dilation. In particular, we show that if the dilation of a map from a fuzzy metric space which is complete in George and Veeramani sense into itself is less than the minimal slope, then the map must have a fixed point. In case that a fuzzy metric space is considered in the sense of Kramosil and Michalek and that the completeness in the sense of Grabiec, the same result holds.

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

LetM be a complete non-compact stable minimal hypersurface in a locally symmetric space N of nonnegative Ricci curvature. We prove that if the integral of square norm of the second fundamental form is finite, i.e., ∫M |A |2dv < ∞, then M must be totally geodesic. (© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

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.

LetMbe a complete Riemannian manifold andNa complete noncompact Riemannian manifold. Letϕ:M→Nbe a surjective harmonic morphism. We prove that ifNadmits a subharmonic function with finite Dirichlet integral which is not harmonic, andϕhas finite energy, thenϕis a constant map. Similarly, iffis a subharmonic function onNwhich is not harmonic and such that|df|is bounded, and if∫M|dϕ|<∞, thenϕis a constant map. We also show that ifNm(m≥3)has at least two ends of infinite volume satisfying the Sobolev inequality or positivity of the first eigenvalue of the Laplacian, then there are no nonconstant surjective harmonic morphisms with finite energy. Forp-harmonic morphisms, similar results hold.

Let M n , n ≥ 3, be a complete oriented immersed minimal hypersurface in Euclidean space R n+1. We show that if the total scalar curvature on M is less than the n/2 power of 1/C s , where C s is the Sobolev constant for M, then there are no L 2 harmonic 1-forms on M. As corollaries, such a minimal hypersurface contains no nontrivial harmonic functions with finite Dirichlet integral and so it has only one end. This implies finally that M is a hyperplane.

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 M n , n≥3, be a complete oriented minimal hypersurface in Euclidean space R n+1 . It is shown that, if the total scalar curvature on M is less than the n/2 power of 1/2C s , where C s is the Sobolev constant for M, and the square norm of the second fundamental form |A| 2 is a L 2 function, then M is a hyperplane.

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 Yamabe invariant of manifolds obtained as connected sums along submanifolds of codimension greater than 2. In particular, given a compact smooth manifold M which does not admit metrics of positive scalar curvature, we prove that the Yamabe invariant of M is an upper bound for the Yamabe invariant of any manifold obtained by performing surgery in M on spheres of codimension greater than 2 ., 14 pages, Latex

We show that given n and D, v > 0, there exists a positive number e = E(n, D, v) > 0 such that if a closed n-manifold M satisfies Ric(M) > -6, diam(M) v, then 7ri (M) is almost abelian.

For the dual operator $s_g'^*$ of the linearization $s_g'$ of the scalar curvature function, it is well-known that if $\ker s_g'^*\neq 0$, then $s_g$ is a non-negative constant. In particular, if the Ricci curvature is not flat, then $ {s_g}/(n-1)$ is an eigenvalue of the Laplacian of the metric $g$. In this work, some variational characterizations were performed for the space $\ker s_g'^*$. To accomplish this task, we introduce a fourth-order elliptic differential operator $\mathcal A$ and a related geometric invariant $\nu$. We prove that $\nu$ vanishes if and only if $\ker s_g'^* \ne 0$, and if the first eigenvalue of the Laplace operator is large compared to its scalar curvature, then $\nu$ is positive and $\ker s_g'^*= 0$. Furthermore, we calculated the lower bound on $\nu$ in the case of $\ker s_g'^* = 0$. We also show that if there exists a function which is $\mathcal A$-superharmonic and the Ricci curvature has a lower bound, then the first non-zero eigenvalue of the Laplace operator has an upper bound., Comment: 25pages no figure