Convergence of Hermitian manifolds and the Type IIB flow

TL;DR

This paper establishes convergence criteria for sequences of Hermitian manifolds and applies these to analyze the behavior and singularities of the Type IIB flow, a geometric flow in complex geometry.

Contribution

It formulates convergence criteria for Hermitian manifolds and applies them to the study of the Type IIB flow, providing new insights into its singularity formation.

Findings

Developed convergence criteria for Hermitian manifolds.

Proved precompactness theorems for the Type IIB flow.

Identified models for singularities in the flow.

Abstract

The Type IIB flow is a flow of conformally balanced complex manifolds introduced by Phong, Picard, and Zhang, about whose singularities little is as yet known. We formulate convergence criteria for the Gromov-Cheeger-Hamilton convergence of sequences of Hermitian manifolds, and apply them to precompactness theorems and the existence of singularity models for the Type IIB flow, in analogy with Hamilton's classic compactness theorems and classification of singularities for the Ricci flow.