2,152 research outputs found
Kahler geometry of toric varieties and extremal metrics
Recently Guillemin gave an explicit combinatorial way of constructing "toric"
Kahler metrics on (symplectic) toric varieties, using only data on the moment
polytope. In this paper, differential geometric properties of these metrics are
investigated using Guillemin's construction. In particular, a nice
combinatorial formula for the scalar curvature is given, and the Euler-Lagrange
condition for such "toric" metrics being extremal (in the sense of Calabi) is
derived. A construction, due to Calabi, of a 1-parameter family of extremal
metrics of non-constant scalar curvature is recast very simply and explicitly.
Finally, a curious combinatorial formula for convex polytopes, that follows
from the relation between the total integral of the scalar curvature and the
wedge product of the first Chern class with a suitable power of the Kahler
class, is presented.Comment: 12 pages, submitted to International Journal of Mathematic
Kahler metrics on toric orbifolds
A theorem of E.Lerman and S.Tolman, generalizing a result of T.Delzant,
states that compact symplectic toric orbifolds are classified by their moment
polytopes, together with a positive integer label attached to each of their
facets. In this paper we use this result, and the existence of "global"
action-angle coordinates, to give an effective parametrization of all
compatible toric complex structures on a compact symplectic toric orbifold, by
means of smooth functions on the corresponding moment polytope. This is
equivalent to parametrizing all toric Kahler metrics and generalizes an
analogous result for toric manifolds.
A simple explicit description of interesting families of extremal Kahler
metrics, arising from recent work of R.Bryant, is given as an application of
the approach in this paper. The fact that in dimension four these metrics are
self-dual and conformally Einstein is also discussed. This gives rise in
particular to a one parameter family of self-dual Einstein metrics connecting
the well known Eguchi-Hanson and Taub-NUT metrics.Comment: 26 pages, 2 figure
Toric Kahler Metrics: Cohomogeneity One Examples of Constant Scalar Curvature in Action-Angle Coordinates
In these notes, after an introduction to toric Kahler geometry, we present
Calabi's family of U(n)-invariant extremal Kahler metrics in symplectic
action-angle coordinates and show that it actually contains, as particular
cases, many interesting cohomogeneity one examples of constant scalar
curvature.Comment: 20 pages, 1 figure, for the proceedings of the XI International
Conference on Geometry, Integrability and Quantization, Varna, Bulgaria, June
5--10, 200
Dynamical convexity and elliptic periodic orbits for Reeb flows
A long-standing conjecture in Hamiltonian Dynamics states that the Reeb flow
of any convex hypersurface in carries an elliptic closed
orbit. Two important contributions toward its proof were given by Ekeland in
1986 and Dell'Antonio-D'Onofrio-Ekeland in 1995 proving this for convex
hypersurfaces satisfying suitable pinching conditions and for antipodal
invariant convex hypersurfaces respectively. In this work we present a
generalization of these results using contact homology and a notion of
dynamical convexity first introduced by Hofer-Wysocki-Zehnder for tight contact
forms on . Applications include geodesic flows under pinching conditions,
magnetic flows and toric contact manifolds.Comment: Version 1: 43 pages. Version 2: revised and improved exposition,
corrected misprints, 44 pages. Version 3: final version, 46 pages, 1 figure,
to appear in Mathematische Annale
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