The Hodge-Dirac operator and Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary

TL;DR

This paper extends spectral functional results of the Hodge-Dirac operator to 4-dimensional manifolds with boundary and proves related Dabrowski-Sitarz-Zalecki type theorems, broadening understanding of geometric analysis in boundary contexts.

Contribution

It generalizes spectral Einstein bilinear functionals and proves Dabrowski-Sitarz-Zalecki theorems for manifolds with boundary, a novel extension of prior work on closed manifolds.

Findings

Spectral Einstein bilinear functionals extended to 4D manifolds with boundary

Proof of Dabrowski-Sitarz-Zalecki type theorems for manifolds with boundary

Enhanced understanding of Hodge-Dirac operator in boundary settings

Abstract

In [10], Dabrowski etc. gave spectral Einstein bilinear functionals of differential forms for the Hodge-Dirac operator $d+\delta$ on an oriented even-dimensional Riemannian manifold. In this paper, we generalize the results of Dabrowski etc. to the cases of 4 dimensional oriented Riemannian manifolds with boundary. Furthermore, we give the proof of Dabrowski-Sitarz-Zalecki type theorems associated with the Hodge-Dirac operator for manifolds with boundary.