Bott-Chern Harmonic Forms on Stein Manifolds

TL;DR

This paper proves a vanishing theorem for certain harmonic forms on Stein manifolds with specific boundedness conditions, advancing understanding of complex geometric analysis on these manifolds.

Contribution

It establishes a new vanishing result for $W^{1,2}$ harmonic forms related to the Bott-Chern Laplacian on $d$-bounded Stein manifolds.

Findings

Vanishing of $W^{1,2}$ harmonic forms on Stein manifolds.

Extension of harmonic form theory to Bott-Chern Laplacian.

Insights into complex geometric analysis on bounded Stein manifolds.

Abstract

Let $M$ be an $n$-dimensional $d$-bounded Stein manifold $M$, i.e., a complex $n$-dimensional manifold $M$ admitting a smooth strictly plurisubharmonic exhaustion $\rho$ and endowed with the K\"ahler metric whose fundamental form is $\omega=i\partial\overline{\partial}\rho$, such that $i\overline{\partial}\rho$ has bounded $L^\infty$ norm. We prove a vanishing result for $W^{1,2}$ harmonic forms with respect to the Bott-Chern Laplacian on $M$.