Abstract
We present a local classification of conformally equivalent but oppositely oriented 4dimensional Kähler metrics which are toric with respect to a common 2torus action. In the generic case, these “ambitoric” structures have an intriguing local geometry depending on a quadratic polynomial q and arbitrary functions A and B of one variable.
We use this description to classify 4dimensional Einstein metrics which are hermitian with respect to both orientations, as well as a class of solutions to the Einstein–Maxwell equations including riemannian analogues of the Plebański–Demiański metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4manifolds, obtained independently by R. Bryant and the first and third authors.
These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kähler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kähler metrics. For both the Einstein–Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kähler metrics on toric 4orbifolds with second Betti number b2=2.
We use this description to classify 4dimensional Einstein metrics which are hermitian with respect to both orientations, as well as a class of solutions to the Einstein–Maxwell equations including riemannian analogues of the Plebański–Demiański metrics. Our classification can be viewed as a riemannian analogue of a result in relativity due to R. Debever, N. Kamran, and R. McLenaghan, and is a natural extension of the classification of selfdual Einstein hermitian 4manifolds, obtained independently by R. Bryant and the first and third authors.
These Einstein metrics are precisely the ambitoric structures with vanishing Bach tensor, and thus have the property that the associated toric Kähler metrics are extremal (in the sense of E. Calabi). Our main results also classify the latter, providing new examples of explicit extremal Kähler metrics. For both the Einstein–Maxwell and the extremal ambitoric structures, A and B are quartic polynomials, but with different conditions on the coefficients. In the sequel to this paper we consider global examples, and use them to resolve the existence problem for extremal Kähler metrics on toric 4orbifolds with second Betti number b2=2.
Original language  English 

Pages (fromto)  109147 
Number of pages  39 
Journal  Journal Fur Die Reine Und Angewandte Mathematik 
Volume  2016 
Issue number  721 
Early online date  19 Aug 2014 
DOIs  
Publication status  Published  1 Dec 2016 
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Profiles

David Calderbank
 Department of Mathematical Sciences  Professor
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching