Biharmonic holomorphic maps and conformally Kahler geometry

TL;DR

This paper explores conditions under which holomorphic maps from almost Hermitian manifolds to (1,2)-symplectic manifolds are biharmonic, extending classical harmonic map results and providing new examples and methods.

Contribution

It introduces conditions on the Lee vector field that ensure holomorphic maps are biharmonic, generalizing the Lichnerowicz theorem and simplifying in locally conformally Kähler manifolds.

Findings

Holomorphic maps satisfy biharmonic conditions under specific Lee vector field constraints

Conditions simplify significantly on locally conformally Kähler manifolds

Constructed examples demonstrate the applicability of the theoretical results

Abstract

We give conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map from this manifold into a (1,2)-symplectic manifold must satisfy the fourth-order condition of being biharmonic, hence generalizing the Lichnerowicz theorem on harmonic maps. These third-order non-linear conditions are shown to greatly simplify on l.c.K. manifolds and construction methods and examples are given in all dimensions.