Biharmonic almost complex structures

Weiyong He

Forum of Mathematics, Sigma, Tome 11 (2023)

Résumé

This project uses methods in geometric analysis to study almost complex manifolds. We introduce the notion of biharmonic almost complex structure on a compact almost Hermitian manifold and study its regularity and existence in dimension four. First, we show that there always exists smooth energy-minimizing biharmonic almost-complex structures for any almost Hermitian four manifold. Then, we study the existence problem where the homotopy class is specified. Given a homotopy class $[\tau ]$ of an almost complex structure, using the fact $\pi _4(S^2)=\mathbb {Z}_2$, there exists a canonical operation p on the homotopy classes satisfying $p^2=\text {id}$ such that $p([\tau ])$ and $[\tau ]$ have the same first Chern class. We prove that there exists an energy-minimizing biharmonic almost complex structure in the companion homotopy classes $[\tau ]$ and $p([\tau ])$. Our results show that, When M is simply connected, there exists an energy-minimizing biharmonic almost complex structure in the homotopy classes with the given first Chern class.

Détail

DOI : 10.1017/fms.2023.21