Diff. Geom, Math. Phys., PDE Seminar: Weiyong He

We study the existence and regularity of harmonic and biharmonic almost complex structures.

Harmonic almost complex structures were introduced by C. Wood in 1990s. Since then there are considerate interest. For harmonic almost complex structure, we prove that the seminal regularity results in the theory of harmonic maps hold similarly in the setting of harmonic almost complex structures. We follow the classical work of Schoen-Uhlenbeck, and recent advance of Cheeger-Naber. A new ingredient is the construction of a comparison almost complex structure used in the regularity, which are different from the classical work of Schoen-Uhlenbeck.

We introduce the notion of biharmonic almost complex structure and study the existence and regularity, in particular in dimension four. Weakly biharmonic almost complex structures are smooth in dimension four. We also prove existence results in a fixed homotopy class and our method relies on a new extension theorem for Sobolev maps (and almost complex structures).