# The harmonic heat flow of almost complex structures

## TL;DR

This paper introduces a harmonic heat flow for almost complex structures compatible with a Riemannian metric, proving long-term existence and convergence to Kähler structures under small energy conditions, and analyzing finite-time singularities.

## Contribution

It defines a tensor-valued harmonic heat flow for almost complex structures and establishes existence, convergence, and singularity results, extending harmonic map heat flow theory.

## Key findings

- Flow exists for all time with small initial energy.
- Flow converges to a Kähler structure.
- Finite time blow-up occurs under certain conditions.

## Abstract

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure $(M, g)$. This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure $J$ has small energy (depending on the norm $|\nabla J|$), then the flow exists for all time and converges to a K\"ahler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no K\"ahler structure in the homotopy class. A main technical tool is a version of monotonicity formula, similar as in the theory of the harmonic map heat flow. We also construct an almost complex structure on a flat four tori with small energy such that the harmonic heat flow blows up at finite time with such an initial data.