Energy minimizing harmonic almost complex structures

Abstract

Consider all compatible almost complex structure with a fixed almost Hermitian metric (M, g) and we study energy-minimizing almost complex structures, with the energy functional \(E(J)=\int _M |\nabla J|^2dv\). C. Wood studied this problem in 1990 s and he named a critical point harmonic almost complex structure, satisfying the equation \([J, \Delta J]=0\). Since then there are considerate interest to study these objects but in general harmonic almost complex structures are not well understood. We introduce the notion of admissible almost complex structure as a natural generalization of smooth almost complex structure to \(W^{1, 2}\) setting. The main goal of the paper is to study the regularity of admissible energy-minimizing almost complex structures on a compact Riemannian (almost Hermitian) manifold (M, g) of even dimension \(m=2n\). Our starting point is to observe that a critical point satisfies the equation \(\Delta J-J\nabla J\nabla J=0\). This is a semi-linear elliptic system of tensor-valued functions which resembles the elliptic system of harmonic maps (vector-valued functions) in many ways. We prove that, all major results for energy-minimizing harmonic maps hold in our setting, notably the seminal work of Schoen–Uhlenbeck and recent improvement of regularity by Cheeger–Naber. We use comparison almost complex structures as was done by Schoen–Uhlenbeck, and finding comparison almost complex structures which satisfies the constraints is a major technical difficulty. This is the main point that differs from the theory of harmonic maps. Our construction of comparison almost complex structures relies on a new input that given an arbitrary almost complex structure, we can construct a unique (canonical) almost complex structure which is compatible with a given metric.