410 research outputs found
On Sasaki-Einstein manifolds in dimension five
We prove the existence of Sasaki-Einstein metrics on certain simply connected
5-manifolds where until now existence was unknown. All of these manifolds have
non-trivial torsion classes. On several of these we show that there are a
countable infinity of deformation classes of Sasaki-Einstein structures.Comment: 18 pages, Exposition was expanded and a reference adde
Energy properness and Sasakian-Einstein metrics
In this paper, we show that the existence of Sasakian-Einstein metrics is
closely related to the properness of corresponding energy functionals. Under
the condition that admitting no nontrivial Hamiltonian holomorphic vector
field, we prove that the existence of Sasakian-Einstein metric implies a
Moser-Trudinger type inequality. At the end of this paper, we also obtain a
Miyaoka-Yau type inequality in Sasakian geometry.Comment: 27 page
Symmetries of the Hamilton–Jacobi equation
We present a detailed discussion of the infinit esimal symmetries of the Hamilton-Jacobi equation (an arbitrary first order partial equation) Our presentation clucidates the role played by the characteristic system in determining the symmetries. We then specialize to the case of a free particle in one space and one time dimension, and study of local Lie group of point transformations locally isomorphie to O(3,2). We show that the separation of variables of the corresponding Hamilton-Jacobi equation n the form of a sum is related to orbits in the Schrödinger subalgebra of c(3,2). The remaining orbits of o(3,2) yield symmetry related solutions which separate in more complicated product forms. Finally some connections with the primordial equation of hydrodynamics (without force terms) are made
Uniqueness and examples of compact toric Sasaki-Einstein metrics
In [11] it was proved that, given a compact toric Sasaki manifold of positive
basic first Chern class and trivial first Chern class of the contact bundle,
one can find a deformed Sasaki structure on which a Sasaki-Einstein metric
exists. In the present paper we first prove the uniqueness of such Einstein
metrics on compact toric Sasaki manifolds modulo the action of the identity
component of the automorphism group for the transverse holomorphic structure,
and secondly remark that the result of [11] implies the existence of compatible
Einstein metrics on all compact Sasaki manifolds obtained from the toric
diagrams with any height, or equivalently on all compact toric Sasaki manifolds
whose cones have flat canonical bundle. We further show that there exists an
infinite family of inequivalent toric Sasaki-Einstein metrics on for each positive integer .Comment: Statements of the results are modifie
Completely integrable relativistic Hamiltonian systems and separation of variables in Hermitian hyperbolic spaces
The Hamilton–Jacobi and Laplace–Beltrami equations on the Hermitian hyperbolic space HH(2) are shown to allow the separation of variables in precisely 12 classes of coordinate systems. The isometry group of this two-complex-dimensional Riemannian space, SU(2,1), has four mutually nonconjugate maximal abelian subgroups. These subgroups are used to construct the separable coordinates explicitly. All of these subgroups are two-dimensional, and this leads to the fact that in each separable coordinate system two of the four variables are ignorable ones. The symmetry reduction of the free HH(2) Hamiltonian by a maximal abelian subgroup of SU(2,1) reduces this Hamiltonian to one defined on an O(2,1) hyperboloid and involving a nontrivial singular potential. Separation of variables on HH(2) and more generally on HH(n) thus provides a new method of generating nontrivial completely integrable relativistic Hamiltonian systems
K\"{a}hler-Einstein metrics on strictly pseudoconvex domains
The metrics of S. Y. Cheng and S.-T. Yau are considered on a strictly
pseudoconvex domains in a complex manifold. Such a manifold carries a complete
K\"{a}hler-Einstein metric if and only if its canonical bundle is positive. We
consider the restricted case in which the CR structure on is
normal. In this case M must be a domain in a resolution of the Sasaki cone over
. We give a condition on a normal CR manifold which it cannot
satisfy if it is a CR infinity of a K\"{a}hler-Einstein manifold. We are able
to mostly determine those normal CR 3-manifolds which can be CR infinities.
Many examples are given of K\"{a}hler-Einstein strictly pseudoconvex manifolds
on bundles and resolutions.Comment: 30 pages, 1 figure, couple corrections, improved a couple example
Randomizing world trade. II. A weighted network analysis
Based on the misleading expectation that weighted network properties always
offer a more complete description than purely topological ones, current
economic models of the International Trade Network (ITN) generally aim at
explaining local weighted properties, not local binary ones. Here we complement
our analysis of the binary projections of the ITN by considering its weighted
representations. We show that, unlike the binary case, all possible weighted
representations of the ITN (directed/undirected, aggregated/disaggregated)
cannot be traced back to local country-specific properties, which are therefore
of limited informativeness. Our two papers show that traditional macroeconomic
approaches systematically fail to capture the key properties of the ITN. In the
binary case, they do not focus on the degree sequence and hence cannot
characterize or replicate higher-order properties. In the weighted case, they
generally focus on the strength sequence, but the knowledge of the latter is
not enough in order to understand or reproduce indirect effects.Comment: See also the companion paper (Part I): arXiv:1103.1243
[physics.soc-ph], published as Phys. Rev. E 84, 046117 (2011
Symmetry and separation of variables for the Hamilton–Jacobi equation W2t −W2x −W2y =0
We present a detailed group theoretical study of the problem of separation of variables for the characteristic equation of the wave equation in one time and two space dimensions. Using the well-known Lie algebra isomorphism between canonical vector fields under the Lie bracket operation and functions (modulo constants) under Poisson brackets, we associate, with each R-separable coordinate system of the equation, an orbit of commuting constants of the motion which are quadratic members of the universal enveloping algebra of the symmetry algebra o (3,2). In this, the first of two papers, we essentially restrict ourselves to those orbits where one of the constants of the motion can be split off, giving rise to a reduced equation with a nontrivial symmetry algebra. Our analysis includes several of the better known two-body problems, including the harmonic oscillator and Kepler problems, as special cases
Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian
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