Title: From conformally Kähler Einstein-Maxwell metrics to extremal Sasaki twins

Abstract: Let (g, ω) be a Kähler metric (and form) on a compact complex manifold (M, J ). We know from the famous work of Yamabe, Trudinger, Aubin, and Schoen that there always exists some positive smooth function f : M → R⁺ such that the metric h = f ⁻²g has constant scalar curvature.

LeBrun and Apostolov-Calderbank-Gauduchon observed that if (M, J, g, ω) is a compact Kähler surface and f is a killing potential, then h is part of a solution to the Riemannian version of the so-called Einstein-Maxwell equations. Such Einstein-Maxwell solutions are called Strongly Hermitian Einstein-Maxwell solutions while h is called a conformally Kähler Einstein-Maxwell metric. LeBrun constructed many examples of such solutions on Hirzebruch surfaces and on the first Hirzebruch surface he discovered an interesting bifurcation: a pair of extra solutions “pop up” when we reach a certain subcone of the Kähler cone. Whenever the cohomology class,  ^ω , is an integer class, these extra solutions turn out to provide non-trivial examples of the so-called extremal Sasaki Twins (introduced by Boyer-Legendre-Huang-TF).

In this talk, which is based on joint work with Boyer, Legendre, and Huang, I will introduce the definitions of twins in the Kähler/Sasaki setting and then exhibit some examples (starting with LeBrun’s) both for dimensions four/five and (if time) higher dimensions.