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Balanced metrics on some Hartogs type domains over bounded symmetric domains  [PDF]
Zhiming Feng,Zhenhan Tu
Mathematics , 2014, DOI: 10.1007/s10455-014-9447-8
Abstract: The definition of balanced metrics was originally given by Donaldson in the case of a compact polarized K\"{a}hler manifold in 2001, who also established the existence of such metrics on any compact projective K\"{a}hler manifold with constant scalar curvature. Currently, the only noncompact manifolds on which balanced metrics are known to exist are homogeneous domains. The generalized Cartan-Hartogs domain $\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ is defined as the Hartogs type domain constructed over the product $\prod_{j=1}^k\Omega_j$ of irreducible bounded symmetric domains $\Omega_j$ $(1\leq j \leq k)$, with the fiber over each point $(z_1,...,z_k)\in \prod_{j=1}^k\Omega_j$ being a ball in $\mathbb{C}^{d_0}$ of the radius $\prod_{j=1}^kN_{\Omega_j}(z_j,\bar{z_j})^{\frac{\mu_j}{2}}$ of the product of positive powers of their generic norms. Any such domain $\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ $(k\geq 2)$ is a bounded nonhomogeneous domain. The purpose of this paper is to obtain necessary and sufficient conditions for the metric $\alpha g(\mu)$ $(\alpha>0)$ on the domain $\big(\prod_{j=1}^k\Omega_j\big)^{{\mathbb{B}}^{d_0}}(\mu)$ to be a balanced metric, where $g(\mu)$ is its canonical metric. As the main contribution of this paper, we obtain the existence of balanced metrics for a class of such bounded nonhomogeneous domains.
Subharmonicity of the variations of K?hler-Einstein metrics on pseudoconvex domains  [PDF]
Young-Jun Choi
Mathematics , 2015,
Abstract: This paper is a sequel to \cite{Choi} in Math. Ann. In that paper we studied the subharmonicity of K\"ahler-Einstein metrics on strongly pseudoconvex domains of dimension greater than or equal to $3$. In this paper, we study the variations K\"ahler-Einstein metrics on bounded strongly pseudoconvex domains of dimension $2$. In addition, we discuss the previous result with general bounded pseudoconvex domain and local triviality of a family of bounded strongly pseudoconvex domains.
Local existence and uniqueness for exterior static vacuum Einstein metrics  [PDF]
Michael T Anderson
Mathematics , 2013,
Abstract: We study solutions to the static vacuum Einstein equations on exterior domains with prescribed metric and mean curvature on the inner boundary. It is proved that for any such boundary data near the standard round boundary data in Euclidean space, there exists a unique AF solution to the static vacuum equations realizing the boundary data, which is close to the standard flat solution.
The equivalent classical metrics on the Cartan-Hartogs Domains  [PDF]
Weiping Yin,An Wang
Mathematics , 2005,
Abstract: In this paper we study the complete invariant metrics on Cartan-Hartogs domains which are the special types of Hua domains. Firstly, we introduce a class of new complete invariant metrics on these domains, and prove that these metrics are equivalent to the Bergman metric. Secondly, the Ricci curvatures under these new metrics are bounded from above and below by the negative constants. Thirdly, we estimate the holomorphic sectional curvatures of the new metrics, we prove that the holomorphic sectional curvatures are bounded from above and below by the negative constants. Finally, by using these new metrics and Yau's Schwarz lemma we prove that the Bergman metric is equivalent to the Einstein-K\"ahler metric. That means the Yau's conjecture is true on Cartan-Hartogs domain.
On analyticity of static vacuum metrics at non-degenerate horizons  [PDF]
Piotr T. Chrusciel
Physics , 2004,
Abstract: We show that static metrics solving vacuum Einstein equations (possibly with a cosmological constant) are one-sided analytic at non-degenerate Killing horizons. We further prove analyticity in a two-sided neighborhood of "bifurcate horizons". It is a pleasure to dedicate this work to Prof. Staruszkiewicz, on the occasion of his 65th birthday.
Dynamical construction of K?hler-Einstein metrics on bounded pseudoconvex domains  [PDF]
Hajime Tsuji
Mathematics , 2013,
Abstract: In this note we shall prove that the complete K\"{a}hler-Einstein volume form on a bounded strongly pseudoconvex domain with $C^{\infty}$-boundary is the normalized limit of a sequence of Bergman kernels.
On scalar curvature rigidity of Vacuum Static Spaces  [PDF]
Jie Qing,Wei Yuan
Mathematics , 2014,
Abstract: In this paper we extend the local scalar curvature rigidity result in [6] to a small domain on general vacuum static spaces, which confirms the interesting dichotomy of local surjectivity and local rigidity about the scalar curvature in general in the light of the paper [10]. We obtain the local scalar curvature rigidity of bounded domains in hyperbolic spaces. We also obtain the global scalar curvature rigidity for conformal deformations of metrics in the domains, where the lapse functions are positive, on vacuum static spaces with positive scalar curvature, and show such domains are maximal, which generalizes the work in [15].
Uniqueness of static vacuum Einstein metrics and the Bartnik quasi-local mass  [PDF]
Michael T. Anderson,Marcus A. Khuri
Mathematics , 2010,
Abstract: We analyse the issue of uniqueness of solutions of the static vacuum Einstein equations with prescribed geometric or Bartnik boundary data. Large classes of examples are constructed where uniqueness fails. We then discuss the implications of this behavior for the Bartnik quasi-local mass. A variational characterization of Bartnik boundary data is also given.
Velocity and velocity bounds in static spherically symmetric metrics  [PDF]
I. Arraut,D. Batic,M. Nowakowski
Physics , 2010, DOI: 10.2478/s11534-010-0147-0
Abstract: We find simple expressions for velocity of massless particles in dependence of the distance $r$ in Schwarzschild coordinates. For massive particles these expressions put an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordstr\"om with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there exists always a region where the massless particle moves with a velocity bigger than the velocity of light in vacuum. In the case of Reissner-Nordstr\"om-de Sitter we completely characterize the radial velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.
The Kaehler-Einstein metric for some Hartogs domains over bounded symmetric domains  [PDF]
An WANG,Weiping YIN,Liyou ZHANG,Guy ROOS
Mathematics , 2005, DOI: 10.1007/s11425-006-0230-6
Abstract: We study the complete K\"{a}hler-Einstein metric of a Hartogs domain $\widetilde {\Omega}$, which is obtained by inflation of an irreducible bounded symmetric domain $\Omega $, using a power $N^{\mu}$ of the generic norm of $\Omega$. The generating function of the K\"{a}hler-Einstein metric satisfies a complex Monge-Amp\`{e}re equation with boundary condition. The domain $\widetilde {\Omega}$ is in general not homogeneous, but it has a subgroup of automorphisms, the orbits of which are parameterized by $X\in\lbrack0,1[$. This allows to reduce the Monge-Amp\`{e}re equation to an ordinary differential equation with limit condition. This equation can be explicitly solved for a special value $\mu_{0}$ of $\mu$, called the critical exponent. We work out the details for the two exceptional symmetric domains. The critical exponent seems also to be relevant for the properties of other invariant metrics like the Bergman metric; a conjecture is stated, which is proved for the exceptional domains.
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