The harmonic heat flow of almost complex structures

Abstract

We define and study the harmonic heat flow for almost complex structures which are compatible with a Riemannian structure ( M , g ) (M, g) . This is a tensor-valued version of harmonic map heat flow. We prove that if the initial almost complex structure J J has small energy (depending on the norm | ∇ J | |\nabla J| ), then the flow exists for all time and converges to a Kähler structure. We also prove that there is a finite time singularity if the initial energy is sufficiently small but there is no Kähler 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.