Variational Structure and Uniqueness of Generalized Kähler–Ricci Solitons

Abstract

Under broad hypotheses we derive a scalar reduction of the generalized Kähler–Ricci soliton system. We realize solutions as critical points of a functional, analogous to the classical Aubin energy, defined on an orbit of the natural Hamiltonian action of diffeomorphisms, thought of as a generalized Kähler class. This functional is convex on a large set of paths in this space, and using this we show rigidity of solitons in their generalized Kähler class. As an application we prove uniqueness of the generalized Kähler–Ricci solitons on Hopf surfaces constructed in Streets and Ustinovskiy [Commun. Pure Appl. Math. 74(9), 1896–1914 (2020)], finishing the classification in complex dimension 2.