Non-positive Scalar Curvature Research Articles

On Einstein warped product space with respect to semi symmetric metric connection

In this paper, we study Einstein warped product space with respect to semi symmetric metric connection. During this study we establish some results on curvature, Ricci and scalar tensors with respect to semi symmetric metric connection and second order semi symmetric metric connection. In the last section, we investigate under what conditions, if $M$ is an Einstein warped space with nonpositive scalar curvature and compact base with respect to semi symmetric metric connection then $M$ is simply a Riemannian product space.

Einstein Poisson warped product space

In this paper, we provide some results on Poisson manifold (M, Π) with contravariant Levi–Civita connection associated to pair (Π, g). We introduce the notion of Einstein Poisson warped product space (M = B × f F, Π, g f ) (where Π = Π1 + Π2). Moreover, we show that if M is an Einstein Poisson warped product space with nonpositive scalar curvature and compact base B, J 1 is a field endomorphism on T*B satisfies , then M is simply a Riemannian Poisson product. For a contravariant Lorentzian Poisson warped space (M = B × f F, g, Π) (where ) one can determine contravariant Einstein equations and the cosmological constant Λ corresponding to the contravariant Einstein equation G = −Λg. Moreover, it is shown that Einstein equation G = −Λg, induces the contravariant Einstein equation with cosmological constant Λ F on fiber space (F, g F , Π F ).

Application of Hyper-generalized Quasi-Einstein Spacetimes in General Relativity

In this paper, we study the hyper-generalized quasi-Einstein (HGQE) warped product spaces with non-positive scalar curvature. This note deals with investigating of some geometric and physical properties of $$(HGQE)_{n}$$ manifolds. Next, we study the general relativistic viscous fluid $$(HGQE)_{4}$$ spacetimes with some physical applications. Lastly, we show the existence of $$(HGQE)_{4}$$ spacetimes by constructing a non-trivial example.

Compact Einstein multiply warped product space with nonpositive scalar curvature

In this paper, we characterize the Einstein multiply warped product space with nonpositive scalar curvature. As a result, it is shown that, if [Formula: see text] is Einstein multiple-warped product spaces with compact base and nonpositive scalar curvature, then [Formula: see text] is simply a Riemannian manifold. Next, we apply our result on Generalized Robertson–Walker space-time and Generalized Friedmann–Robertson–Walker space-time.

A geometric description for simple and damped harmonic oscillators

In this work we consider the Riemannian geometry associated with the differential equations of one dimensional simple and damped linear harmonic oscillators. We show that the sectional curvatures are completely determined by the oscillation frequency and the friction coefficient and these physical constants can be thought as obstructions for the manifold to be flat. Moreover, equations of simple and damped harmonic oscillators describe nonisomorphic solvable Lie groups with nonpositive scalar curvature.

Yamabe Flow on Non-compact Manifolds with Unbounded Initial Curvature

We prove global existence of Yamabe flows on non-compact manifolds $M$ of dimension $m\geq3$ under the assumption that the initial metric $g_0=u_0g_M$ is conformally equivalent to a complete background metric $g_M$ of bounded, non-positive scalar curvature and positive Yamabe invariant with conformal factor $u_0$ bounded from above and below. We do not require initial curvature bounds. In particular, the scalar curvature of $(M,g_0)$ can be unbounded from above and below without growth condition.

Complete Yamabe solitons with finite total scalar curvature

In this paper, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive Ricci curvature are Ricci flat. Moreover, under certain pinching condition for Ricci curvature, we show that steady or shrinking complete gradient Yamabe solitons with finite total scalar curvature and non-positive scalar curvature are Ricci flat.

On compact Einstein doubly warped product manifolds

In this paper, the non-existence of connected, compact Einstein doubly warped product semi-Riemannian manifold with non-positive scalar curvature is proved. It is also shown that there does not exist non-trivial connected Einstein doubly warped product semi-Riemannian manifold with compact base $B$ or fibre $F$.

A remark on the mixed scalar curvature of a manifold with two orthogonal totally umbilical distributions

Abstract We prove a Liouville-type theorem for two orthogonal complementary totally umbilical distributions on a complete Riemannian manifold with non-positive mixed scalar curvature. This is applied to some special types of complete doubly twisted and warped products of Riemannian manifolds.

Yamabe classification and prescribed scalar curvature in the asymptotically Euclidean setting

We prove a necessary and sufficient condition for an asymptotically Euclidean manifold to be conformally related to one with specified nonpositive scalar curvature: the zero set of the desired scalar curvature must have a positive Yamabe invariant, as defined in the article. We show additionally how the sign of the Yamabe invariant of a measurable set can be computed from the sign of certain generalized weighted eigenvalues of the conformal Laplacian. Using the prescribed scalar curvature result we give a characterization of the Yamabe classes of asymptotically Euclidean manifolds. We also show that the Yamabe class of an asymptotically Euclidean manifold is the same as the Yamabe class of its conformal compactification.

On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature

On Compact Super Quasi-Einstein Warped Product with Nonpositive Scalar Curvature

Einstein manifolds with finite Lp-norm of the Weyl curvature

Let (Mn,g)(n≥4) be an n-dimensional complete Einstein manifold. Denote by W the Weyl curvature tensor of M. We prove that (Mn,g) is isometric to a spherical space form if (Mn,g) has positive scalar curvature and unit volume, and the Lp(p≥n2)-norm of W is pinched in [0,C), where C is an explicit positive constant depending only on n, p and S, which improves the isolation theorems given by [24,14,17].This paper also states that W goes to zero uniformly at infinity if for p≥n2, the Lp-norm of W of M with non-positive scalar curvature and positive Yamabe constant is finite. Assume that M has negative scalar curvature and the Lα-norm of W is finite. As application, we prove that M is a hyperbolic space form if the Lp-norm of W is sufficiently small, which generalizes an Ln2-norm of W pinching theorem in [19].

On the construction of gradient Ricci soliton warped product

In this paper we show that an expanding or steady gradient Ricci soliton warped product Bn×fFm, m>1, whose warping function f reaches both maximum and minimum must be a Riemannian product. Moreover, we present a necessary and sufficient condition for constructing a gradient Ricci soliton warped product. As an application, we present a class of expanding Ricci soliton warped product having as a fiber an Einstein manifold with non-positive scalar curvature. We also discuss some obstructions to this construction, especially in the case when the base of the warped product is compact.

Brown–York Mass and Compactly Supported Conformal Deformations of Scalar Curvature

In this article, we find a connection between Brown–York mass and the first Dirichlet eigenvalue of a Schrodinger type operator. In particular, we prove a local positive mass type theorem for metrics conformal to the background one with suitable presumptions. As applications, we investigate compactly supported conformal deformations which either increase or decrease scalar curvature. We find local conformal rigidity phenomena occur in both cases for small domains and as for manifolds with nonpositive scalar curvature it is even more rigid in particular. On the other hand, such deformations exist for closed or a type of non-compact manifolds with positive scalar curvature. These results together give an answer to a question that arises naturally in (Corvino in Commun Math Phys 214:137–189, 2000; Lohkamp in Math Ann 313:385–407, 1999).

Monotonicity of the first Dirichlet eigenvalue of the Laplacian on manifolds of non-positive curvature

Let (M,g) be a complete Riemannian manifold with nonpositive scalar curvature, let ⊂ M be a suitable domain, and let λ() be the first Dirichlet eigenvalue of the Laplace-Beltrami operator on . We prove several bounds for the rate of decrease of λ() as increases, and a result comparing the rate of decrease of λ before and after

Hessian of the natural Hermitian form on twistor spaces

We compute the hessian of the natural Hermitian form successively on the Calabi family of a hyperk\"ahler manifold, on the twistor space of a 4-dimensional anti-self-dual Riemannian manifold and on the twistor space of a quaternionic K\"ahler manifold. We show a strong convexity property of the cycle space of twistor lines on the Calabi family of a hyperk\"ahler manifold. We also prove convexity properties of the 1-cycle space of the twistor space of a 4-dimensional anti-self-dual Einstein manifold of non-positive scalar curvature and of the 1-cycle space of the twistor space of a quaternionic K\"ahler manifold of non-positive scalar curvature. We check that no non-K\"ahler strong KT manifold occurs as such a twistor space.

Examples of Einstein manifolds in odd dimensions

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kahler–Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain 3-sphere bundles over a Fano Kahler–Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane.

Asymptotic Curvature of Moduli Spaces for Calabi–Yau Threefolds

Motivated by the classical statements of Mirror Symmetry, we study certain Kahler metrics on the complexified Kahler cone of a Calabi–Yau threefold, conjecturally corresponding to approximations to the Weil–Petersson metric near large complex structure limit for the mirror. In particular, the naturally defined Riemannian metric (defined via cup-product) on a level set of the Kahler cone is seen to be analogous to a slice of the Weil–Petersson metric near large complex structure limit. This enables us to give counterexamples to a conjecture of Ooguri and Vafa that the Weil–Petersson metric has non-positive scalar curvature in some neighborhood of the large complex structure limit point.

Liouville type theorems for p-harmonic maps

Let M be a complete Riemannian manifold and let N be a Riemannian manifold of non-positive sectional curvature. Assume that Ric M ⩾ − 4 ( p − 1 ) p 2 μ 0 at all x ∈ M and > − 4 ( p − 1 ) p 2 μ 0 at some point x 0 ∈ M , where μ 0 > 0 is the least eigenvalue of the Laplacian acting on L 2 -functions on M. Let 2 ⩽ q ⩽ p . Then any q-harmonic map ϕ : M → N of finite q-energy is constant. Moreover, if N is a Riemannian manifold of non-positive scalar curvature, then any q-harmonic morphism ϕ : M → N of finite q-energy is constant.

Classification and Liouville type theorems for p-harmonic morphisms

We prove firstly the classification theorem for p-harmonic morphisms between Euclidean domains. Secondly, we show that if \(\phi: M\to N\) is a p-harmonic morphism (p ≥ 2) from a complete Riemannian manifold M of nonnegative Ricci curvature into a Riemannian manifold N of non-positive scalar curvature such that the Lq-energy is finite, then \(\phi\) is constant, which improve the corresponding result due to G. Choi, G. Yun in (Geometriae Dedicata 101 (2003), 53–59).