AMBITORIC GEOMETRY II: EXTREMAL TORIC SURFACES AND EINSTEIN 4-ORBIFOLDS

作者：Apostolov Vestislav^*; Calderbank David M J; Gauduchon Paul

来源：Annales Scientifiques de l'Ecole Normale Superieure, 2015, 48(5): 1075-1112.

DOI：10.24033/asens.2266

We provide an explicit resolution of the existence problem for extremal Kahler metrics on toric 4-orbifolds M with second Betti number b(2) (M) = 2. More precisely we show that M admits such a metric if and only if its rational Delzant polytope (which is a labelled quadrilateral) is K-polystable in the relative, toric sense (as studied by S. Donaldson, E. Legendre, G. Szekelyhidi et al.). Furthermore, in this case, the extremal Kahler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kahler metric, which turns out to be extremal as well. These results provide a computational test for the K-stability of labelled quadrilaterals.
Extremal ambitoric structures were classified locally in Part I of this work, but herein we only use the straightforward fact that explicit Kahler metrics obtained there are extremal, and the identification of Bach-fiat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-fiat Kahler orbifolds, including examples which are globally conformally Einstein, and examples which are conformal to complete smooth Einstein metrics on an open subset, thus extending the work of many authors.

• 出版日期2015-10