[Paper Review] Energy minimizing harmonic almost complex structures

Weiyong He|arXiv (Cornell University)|Jul 29, 2019

Geometric Analysis and Curvature Flows43 references4 citations

TL;DR

This paper introduces admissible almost complex structures in the $W^{1,2}$ Sobolev setting to study energy-minimizing harmonic almost complex structures on compact almost Hermitian manifolds. By reformulating the Euler-Lagrange equation as a semi-linear elliptic system $\Delta J - J\nabla_p J \nabla_p J = 0$, the author establishes full regularity via $\epsilon$-regularity and quantitative stratification, proving that energy-minimizing admissible almost complex structures are smooth, even on non-complex manifolds like $S^4$. The results extend Schoen-Uhlenbeck's regularity theory to this non-tensorial, metric-constrained setting.

ABSTRACT

We study the existence and regularity of energy-minimizing harmonic almost complex structures. We have proved results similar to the theory of harmonic maps, notably the classical results of Schoen-Uhlenbeck and recent advance by Cheeger-Naber.

Motivation & Objective

• To define and study admissible almost complex structures in the $W^{1,2}$ Sobolev space as a generalization of smooth almost complex structures.
• To establish the existence and compactness of energy-minimizing admissible almost complex structures on compact Riemannian manifolds, including $S^4$, which admits no smooth almost complex structure.
• To prove that energy-minimizing admissible almost complex structures are smooth, extending regularity theory beyond harmonic maps.

Proposed method

• Introduce admissible almost complex structures as $W^{1,2}$ limits of smooth compatible almost complex structures, preserving compatibility with the background metric.
• Reformulate the harmonic almost complex structure equation as $\Delta J - J\nabla_p J \nabla_p J = 0$, a semi-linear elliptic system resembling harmonic map equations.
• Use comparison almost complex structures constructed via a canonical projection to a metric-compatible almost complex structure, overcoming constraints from the background metric.
• Apply $\epsilon$-regularity and quantitative stratification techniques inspired by Schoen-Uhlenbeck and Cheeger-Naber to control singular sets.
• Use difference quotients and weighted $L^p$ estimates to prove $\partial J \in L^p_{\text{loc}}$ for all $p < \infty$, enabling bootstrapping to smoothness.
• Leverage the Bochner formula and monotonicity formula for energy-minimizing systems to control growth and singularities.

Experimental results

Research questions

• RQ1 Can energy-minimizing almost complex structures be regularized in the $W^{1,2}$ setting, even when no smooth compatible structure exists?
• RQ2 How does the background Riemannian metric constrain the regularity theory of harmonic almost complex structures compared to harmonic maps?
• RQ3 To what extent do $\epsilon$-regularity and quantitative stratification apply to energy-minimizing almost complex structures?
• RQ4 Can the singular set of an energy-minimizing admissible almost complex structure be stratified and shown to have finite codimension?
• RQ5 Is a weakly harmonic almost complex structure with $L^2$ gradient necessarily smooth?

Key findings

• Energy-minimizing admissible almost complex structures exist on any compact Riemannian manifold of even dimension, including $S^4$, and form a compact set in the $W^{1,2}$ topology.
• All major regularity results for energy-minimizing harmonic maps—such as $\epsilon$-regularity, stratification of the singular set, and the regularity scale—extend to energy-minimizing admissible almost complex structures.
• The $L^p$ integrability of $\partial J$ for all $p \in (1, \infty)$ is established via an induction argument on difference quotients and weighted estimates.
• The Euler-Lagrange equation $\Delta J - J\nabla_p J \nabla_p J = 0$ is a semi-linear elliptic system that supports full regularity via bootstrapping from $W^{2,p}$ to $C^{1,\alpha}$ and then to smoothness.
• The canonical construction of a metric-compatible almost complex structure from an arbitrary one enables the construction of comparison structures essential for the $\epsilon$-regularity proof.
• The singular set of an energy-minimizing admissible almost complex structure has Hausdorff codimension at least 4, analogous to harmonic maps, and is stratified by regularity scale.