An $L^2$-$\partial\overline\partial$-Lemma on a class of complete K\"ahler manifolds

TL;DR

This paper extends the classical $ ext{d} ext{d}^c$-lemma to certain complete K"ahler manifolds by establishing an $L^2$-$ ext{d} ext{d}^c$-lemma under spectral gap conditions on the Hodge Laplacian.

Contribution

It proves an $L^2$-$ ext{d} ext{d}^c$-lemma for complete K"ahler manifolds with spectral gap assumptions, generalizing the classical lemma from compact cases.

Findings

Establishes an $L^2$-$ ext{d} ext{d}^c$-lemma under spectral gap conditions.

Generalizes the classical $ ext{d} ext{d}^c$-lemma to non-compact K"ahler manifolds.

Provides conditions for the Hodge Laplacian's spectrum to ensure the lemma holds.

Abstract

We prove an $L^2$-$\partial\overline\partial$-Lemma involving smooth square integrable forms on complete K\"ahler 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 K\"ahler manifolds.