49 research outputs found
Perelman's entropy for some families of canonical metrics
We numerically calculate Perelmanâs entropy for a variety of canonical metrics on CP1-bundles over products of Fano KĂ€hler-Einstein manifolds. The metrics investigated are Einstein metrics, KĂ€hler-Ricci solitons and quasi-Einstein metrics. The calculation of the entropy allows a rough picture of how the Ricci flow behaves on each of the manifolds in question
Special K\"ahler-Ricci potentials on compact K\"ahler manifolds
A special K\"ahler-Ricci potential on a K\"ahler manifold is any nonconstant
function such that is a Killing vector field
and, at every point with , all nonzero tangent vectors orthogonal
to and are eigenvectors of both and
the Ricci tensor. For instance, this is always the case if is a
nonconstant function on a K\"ahler manifold of complex
dimension and the metric , defined wherever , is Einstein. (When such exists, may be called {\it
almost-everywhere conformally Einstein}.) We provide a complete classification
of compact K\"ahler manifolds with special K\"ahler-Ricci potentials and use it
to prove a structure theorem for compact K\"ahler manifolds of any complex
dimension which are almost-everywhere conformally Einstein.Comment: 45 pages, AMSTeX, submitted to Journal f\"ur die reine und angewandte
Mathemati
Future geodesic completeness of some spatially homogeneous solutions of the vacuum Einstein equations in higher dimensions
It is known that all spatially homogeneous solutions of the vacuum Einstein
equations in four dimensions which exist for an infinite proper time towards
the future are future geodesically complete. This paper investigates whether
the analogous statement holds in higher dimensions. A positive answer to this
question is obtained for a large class of models which can be studied with the
help of Kaluza-Klein reduction to solutions of the Einstein-scalar field
equations in four dimensions. The proof of this result makes use of a criterion
for geodesic completeness which is applicable to more general spatially
homogeneous models.Comment: 18 page
Compact conformally Kahler Einstein-Weyl manifolds
We give a classification of compact conformally Kahler Einstein-Weyl
manifolds whose Ricci tensor is hermitian.Comment: 11 page
Kahler manifolds with quasi-constant holomorphic curvature
The aim of this paper is to classify compact Kahler manifolds with
quasi-constant holomorphic sectional curvature.Comment: 18 page
How to find the holonomy algebra of a Lorentzian manifold
Manifolds with exceptional holonomy play an important role in string theory,
supergravity and M-theory. It is explained how one can find the holonomy
algebra of an arbitrary Riemannian or Lorentzian manifold. Using the de~Rham
and Wu decompositions, this problem is reduced to the case of locally
indecomposable manifolds. In the case of locally indecomposable Riemannian
manifolds, it is known that the holonomy algebra can be found from the analysis
of special geometric structures on the manifold. If the holonomy algebra
of a locally indecomposable
Lorentzian manifold of dimension is different from
, then it is contained in the similitude algebra
. There are 4 types of such holonomy algebras. Criterion
how to find the type of are given, and special geometric
structures corresponding to each type are described. To each
there is a canonically associated subalgebra
. An algorithm how to find
is provided.Comment: 15 pages; the final versio
Lorentzian manifolds and scalar curvature invariants
We discuss (arbitrary-dimensional) Lorentzian manifolds and the scalar
polynomial curvature invariants constructed from the Riemann tensor and its
covariant derivatives. Recently, we have shown that in four dimensions a
Lorentzian spacetime metric is either -non-degenerate, and hence
locally characterized by its scalar polynomial curvature invariants, or is a
degenerate Kundt spacetime. We present a number of results that generalize
these results to higher dimensions and discuss their consequences and potential
physical applications.Comment: submitted to CQ
Hermitian Yang-Mills instantons on resolutions of Calabi-Yau cones
We study the construction of Hermitian Yang-Mills instantons over resolutions
of Calabi-Yau cones of arbitrary dimension. In particular, in d complex
dimensions, we present an infinite family, parametrised by an integer k and a
continuous modulus, of SU(d) instantons. A detailed study of their properties,
including the computation of the instanton numbers is provided. We also explain
how they can be used in the construction of heterotic non-Kahler
compactifications.Comment: 20 pages, 1 figure; typos corrected, section 3.1 expande
Remarks on symplectic twistor spaces
We consider some classical fibre bundles furnished with almost complex
structures of twistor type, deduce their integrability in some cases and study
\textit{self-holomorphic} sections of a \textit{symplectic} twistor space. With
these we define a moduli space of -compatible complex structures. We
recall the theory of flag manifolds in order to study the Siegel domain and
other domains alike, which is the fibre of the referred twistor space. Finally
the structure equations for the twistor of a Riemann surface with the canonical
symplectic-metric connection are deduced, based on a given conformal coordinate
on the surface. We then relate with the moduli space defined previously.Comment: 20 pages, title changed since v2, accepted in AMPA toda
Compact Einstein Spaces based on Quaternionic K\"ahler Manifolds
We investigate the Einstein equation with a positive cosmological constant
for -dimensional metrics on bundles over Quaternionic K\"ahler base
manifolds whose fibers are 4-dimensional Bianchi IX manifolds. The Einstein
equations are reduced to a set of non-linear ordinary differential equations.
We numerically find inhomogeneous compact Einstein spaces with orbifold
singularity.Comment: LaTeX 28 pages, 5 eps figure