The BFK-gluing formula and the curvature tensors on a 2-dimensional compact hypersurface

TL;DR

This paper derives explicit formulas for the coefficients of a polynomial in the BFK-gluing formula and heat trace asymptotics, relating them to curvature tensors of a 2D hypersurface in a manifold.

Contribution

It provides new explicit expressions for polynomial coefficients in the BFK-gluing formula using scalar and principal curvatures of 2D hypersurfaces.

Findings

Coefficients expressed in terms of scalar and principal curvatures.

Connections established between geometric data and spectral invariants.

Enhanced understanding of the geometric aspects of the BFK-gluing formula.

Abstract

In the proof of the BFK-gluing formula for zeta-determinants of Laplacians there appears a real polynomial whose constant term is an important ingredient in the gluing formula. This polynomial is determined by geometric data on an arbitrarily small collar neighborhood of a cutting hypersurface. In this paper we express the coefficients of this polynomial in terms of the scalar and principal curvatures of the cutting hypersurface embedded in the manifold when this hypersurface is 2-dimensional. Similarly, we express some coefficients of the heat trace asymptotics of the Dirichlet-to-Neumann operator in terms of the scalar and principal curvatures of the cutting hypersurface.