Spaces of Spheres, Duality Structures, and The Finite Volume Laplacian

Abstract

The main focus of this work is the Kirchhoff determinant of the finite volume Laplacian. The finite volume Laplacian is a type of weighted graph Laplacian which may be used to approximate the smooth Laplacian on a manifold, and also arises when studying variations of curvature quantities for discrete conformal structures in dimensions two and three. We consider the finite volume Laplacian on a single simplex $T$, and relate its Kirchhoff determinant separately to the signed volumes of two other simplices, $T^\sharp$ and $T^\flat$, constructed from the geometry of $T$ as well as a chosen duality structure on $T$ which determines the finite volume Laplacian. A decomposition of the volume of $T^\sharp$ is presented. The construction and results for $T^\flat$ are apparently novel. A duality structure may be understood as an assignment of centers to each subsimplex in a triangulation by orthogonal projection of chosen top-dimensional subsimplex centers, subject to the constraint that centers assigned to shared faces are consistent. Signed dual cells which are determined by a duality structure may be pieced together together to obtain the original simplex $T$, and their structure plays a key role in defining and relating $T^\sharp$ and $T^\flat$. Ideas from linear algebra of spheres are used to motivate the construction of $T^\flat$ and prove some properties about it. A formula for the signed volume of a pedal simplex is also derived.