42 research outputs found
Perelman's entropy for some families of canonical metrics
We numerically calculate Perelman’s entropy for a variety of canonical metrics on CP1-bundles over products of Fano Kähler-Einstein manifolds. The metrics investigated are Einstein metrics, Kähler-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
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
Ricci-flat Metrics with U(1) Action and the Dirichlet Boundary-value Problem in Riemannian Quantum Gravity and Isoperimetric Inequalities
The Dirichlet boundary-value problem and isoperimetric inequalities for
positive definite regular solutions of the vacuum Einstein equations are
studied in arbitrary dimensions for the class of metrics with boundaries
admitting a U(1) action. We show that in the case of non-trivial bundles
Taub-Bolt infillings are double-valued whereas Taub-Nut and Eguchi-Hanson
infillings are unique. In the case of trivial bundles, there are two
Schwarzschild infillings in arbitrary dimensions. The condition of whether a
particular type of filling in is possible can be expressed as a limitation on
squashing through a functional dependence on dimension in each case. The case
of the Eguchi-Hanson metric is solved in arbitrary dimension. The Taub-Nut and
the Taub-Bolt are solved in four dimensions and methods for arbitrary dimension
are delineated. For the case of Schwarzschild, analytic formulae for the two
infilling black hole masses in arbitrary dimension have been obtained. This
should facilitate the study of black hole dynamics/thermodynamics in higher
dimensions. We found that all infilling solutions are convex. Thus convexity of
the boundary does not guarantee uniqueness of the infilling. Isoperimetric
inequalities involving the volume of the boundary and the volume of the
infilling solutions are then investigated. In particular, the analogues of
Minkowski's celebrated inequality in flat space are found and discussed
providing insight into the geometric nature of these Ricci-flat spaces.Comment: 40 pages, 3 figure