The infimum of the Nijenhuis energy

TL;DR

This paper demonstrates that on certain symplectic manifolds, one can construct a sequence of compatible almost complex structures with Nijenhuis energy approaching zero, using neck-stretching techniques around Donaldson hypersurfaces.

Contribution

It introduces a method to produce sequences of compatible almost complex structures with vanishing Nijenhuis energy on rational symplectic manifolds.

Findings

Existence of sequences with Nijenhuis energy tending to zero

Use of neck-stretching around Donaldson hypersurfaces

Applicable to symplectic manifolds with rational cohomology classes

Abstract

We prove that on any symplectic manifold whose symplectic form represents a rational cohomology class there exists a sequence of compatible almost complex structures whose Nijenhuis energy (the $L^2$-norm of the Nijenhuis tensor) tends to zero. The sequence is obtained by stretching the neck around a Donaldson hypersurface.