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

We prove an $L^2$-$\mathrm{\partial }\bar{\mathrm{\partial }}$-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 $\mathrm{\partial }\bar{\mathrm{\partial }}$-Lemma on compact Kähler manifolds.