Polyharmonic Almost Complex Structures

TL;DR

This paper investigates the existence and smoothness of polyharmonic almost complex structures on compact manifolds, proving regularity results including smoothness in four dimensions for biharmonic cases.

Contribution

It establishes a general regularity theorem for semilinear elliptic systems with critical growth, specifically proving smoothness of weakly biharmonic almost complex structures in dimension four.

Findings

Weakly biharmonic almost complex structures are smooth in dimension four.

Established a regularity theorem for semilinear systems with critical growth.

Proved existence and regularity results for polyharmonic almost complex structures.

Abstract

In this paper we consider the existence and regularity of weakly polyharmonic almost complex structures on a compact almost Hermitian manifold $M^{2m}$. Such objects satisfy the elliptic system weakly $[J, \Delta^m J]=0$. We prove a very general regularity theorem for semilinear systems in critical dimensions (with \emph{critical growth nonlinearities}). In particular we prove that weakly biharmonic almost complex structures are smooth in dimension four.