Ambitoric geometry II: Extremal toric surfaces and Einstein 4-orbifolds

TL;DR

This paper characterizes when certain toric 4-orbifolds admit extremal Kähler metrics, linking geometric stability conditions to explicit metric constructions, and identifies infinite families of conformally Einstein examples.

Contribution

It provides a complete criterion for the existence of extremal Kähler metrics on toric 4-orbifolds with b2=2 based on K-stability of labelled quadrilaterals, and constructs new conformally Einstein metrics.

Findings

Characterization of extremal Kähler metrics via K-stability.

Explicit construction of ambitoric extremal metrics.

Infinite families of conformally Einstein toric orbifolds.

Abstract

We provide an explicit resolution of the existence problem for extremal Kaehler metrics on toric 4-orbifolds M with second Betti number b2(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 Kaehler metric is ambitoric, i.e., compatible with a conformally equivalent, oppositely oriented toric Kaehler 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 Kaehler metrics obtained there are extremal, and the identification of Bach-flat (conformally Einstein) examples among them. Using our global results, the latter yield countably infinite families of compact toric Bach-flat Kaehler 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.