Variational structure and uniqueness of generalized K\"ahler-Ricci solitons

TL;DR

This paper establishes a scalar reduction for generalized K"ahler-Ricci solitons, introduces a convex functional framework, and proves the uniqueness of such solitons on Hopf surfaces, completing their classification in complex dimension two.

Contribution

It provides a scalar reduction of the generalized K"ahler-Ricci soliton system and proves the uniqueness of these solitons on Hopf surfaces, advancing the classification in complex dimension two.

Findings

Scalar reduction of the soliton system

Convexity of the functional on Hamiltonian paths

Uniqueness of solitons on Hopf surfaces

Abstract

Under broad hypotheses we derive a scalar reduction of the generalized K\"ahler-Ricci soliton system. We realize solutions as critical points of a functional analogous to the classical Aubin energy defined on the orbit of a natural Hamiltonian action of diffeomorphisms, thought of as a generalized K\"ahler 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 K\"ahler class. As an application we prove uniqueness of the generalized K\"ahler-Ricci solitons on Hopf surfaces constructed in arXiv:1907.03819, finishing the classification in complex dimension $2$.