$L^2$-transverse Killing forms on a transverse Kähler foliation

Abstract

It is well-known that on a compact Riemannian manifold with a transverse Kähler foliation of codimension $q=2m$, any transverse Killing r-form $(r\ge 2)$ is parallel (Jung and Jung in Bull Korean Math Soc 49:445–454, 2012). In this article, we study the parallelness of $L^2$-transverse Killing forms on a complete foliated Riemannian manifold M with a transverse Kähler foliation ${\mathcal {F}}$. Precisely, if all leaves of ${\mathcal {F}}$ are compact and the mean curvature form is transversally holomorphic, coclosed and bounded, then all $L^2$-transverse Killing r-forms $(r\ge 2)$ are parallel. In addition, if the volume of M is infinite, then all $L^2$-transverse Killing r-forms $(r\ge 2)$ are trivial.