Almost complex manifolds with small Nijenhuis tensor

Luis Fernandez, Tobias Shin, Scott O. Wilson

Published 2021-03-10Version 1

We give several explicit examples of compact manifolds with a $1$-parameter family of almost complex structures having arbitrarily small Nijenhuis tensor in the $C^0$-norm. The $4$ dimensional examples possess no complex structure, whereas the dimension $6$ examples do not possess left invariant complex structures, and whether they possess complex structures appears to be unknown.