Toric Anti-self-dual Einstein Metrics via Complex Geometry
نویسنده
چکیده
Using the twistor correspondence, we give a classification of toric anti-self-dual Einstein metrics: each such metric is essentially determined by an odd holomorphic function. This explains how the Einstein metrics fit into the classification of general toric anti-self-dual metrics given in an earlier paper [7]. The results complement the work of Calderbank–Pedersen [6], who describe where the Einstein metrics appear amongst the Joyce spaces, leading to a different classification. Taking the twistor transform of our result gives a new proof of their theorem.
منابع مشابه
Self-dual metrics on toric 4-manifolds: Extending the Joyce construction
Toric geometry studies manifolds M2n acted on effectively by a torus of half their dimension, T . Joyce shows that for such a 4-manifold sufficient conditions for a conformal class of metrics on the free part of the action to be self-dual can be given by a pair of linear ODEs and gives criteria for a metric in this class to extend to the degenerate orbits. Joyce and Calderbank-Pedersen use this...
متن کاملToric Anti-self-dual 4-manifolds via Complex Geometry
Using the twistor correspondence, this article gives a oneto-one correspondence between germs of toric anti-self-dual conformal classes and certain holomorphic data determined by the induced action on twistor space. Recovering the metric from the holomorphic data leads to the classical problem of prescribing the Čech coboundary of 0-cochains on an elliptic curve covered by two annuli. The class...
متن کاملStony Brook University
of the Dissertation Self-Dual Metrics on 4-Manifolds by Mustafa Kalafat Doctor of Philosophy in Mathematics Stony Brook University 2007 Under a vanishing hypothesis, Donaldson and Friedman proved that the connected sum of two self-dual Riemannian 4-Manifolds is again self-dual. Here we prove that the same result can be extended over to the positive scalar curvature case. Secondly we give an exa...
متن کاملToric Self-dual Einstein Metrics as Quotients
We use the quaternion Kähler reduction technique to study old and new selfdual Einstein metrics of negative scalar curvature with at least a two-dimensional isometry group, and relate the quotient construction to the hyperbolic eigenfunction Ansatz. We focus in particular on the (semi-)quaternion Kähler quotients of (semi-)quaternion Kähler hyperboloids, analysing the completeness and topology,...
متن کاملToric Geometry of Convex Quadrilaterals
We provide an explicit resolution of the Abreu equation on convex labeled quadrilaterals. This confirms a conjecture of Donaldson in this particular case and implies a complete classification of the explicit toric Kähler– Einstein and toric Sasaki–Einstein metrics constructed in [6, 23, 14]. As a byproduct, we obtain a wealth of extremal toric (complex) orbi-surfaces, including Kähler–Einstein ...
متن کامل