An $L^2$-$\partial \overline{\partial }$-Lemma on a class of complete Kähler manifolds

Abstract

We prove an $L^2$-$\partial \overline{\partial }$-Lemma involving smooth square integrable forms on complete Kähler manifolds, provided that the unique self-adjoint extension of the Hodge Laplacian on the Hilbert space of $L^2$-forms has a gap in its spectrum near zero. This generalises the classical $\partial \overline{\partial }$-Lemma on compact Kähler manifolds.

15 pages, comments are welcome!