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.