Biharmonic holomorphic maps and conformally Kähler geometry
The aim of this article is to provide a method for constructing biharmonic maps from holomorphic maps. Let \(\phi\) be a map between two Riemannian manifolds, then its tension field is given by \(\tau(\phi):=\text{trace}\nabla d\phi\). Biharmonic maps are critical points of the \(L^2\)-norm of the tension field, the corresponding Euler-Lagrange equation is a fourth order partial differential equation. Motivated by the Lichnerowicz theorem for harmonic maps, the authors derive conditions on the Lee vector field of an almost Hermitian manifold such that any holomorphic map into a \((1,2)\)-symplectic manifold is biharmonic. These conditions turn out to be non-linear and of third order. However, it is shown that the latter simplify on locally conformal Kähler manifolds. In addition, the authors provide construction methods and give examples in arbitrary dimensions.
scientific article; zbMATH DE number 6361308
scientific article