Classification of solitons for pluriclosed flow on complex surfaces

TL;DR

This paper classifies compact solitons for the pluriclosed flow on complex surfaces, showing most are Kähler except certain Hopf surfaces, and constructs explicit steady solitons on these surfaces.

Contribution

It provides a complete classification of solitons on complex surfaces and constructs explicit examples on Hopf surfaces, expanding understanding of pluriclosed flow solutions.

Findings

Most solitons are on Kähler surfaces.

Steady solitons exist on minimal Hopf surfaces.

Explicit steady solitons are constructed on class 1 Hopf surfaces.

Abstract

We give a classification of compact solitons for the pluriclosed flow on complex surfaces. First, by exploiting results from the Kodaira classification of surfaces, we show that the complex surface underlying a soliton must be K\"ahler except for the possibility of steady solitons on minimal Hopf surfaces. Then, we construct steady solitons on all class $1$ Hopf surfaces by exploiting a natural symmetry ansatz.