Weighted renormalized volume coefficients

TL;DR

This paper introduces weighted renormalized volume coefficients, proves their variational property, and expresses them as polynomials involving weighted geometric tensors, advancing the understanding of geometric invariants.

Contribution

It defines weighted renormalized volume coefficients and demonstrates their polynomial expression in terms of weighted geometric tensors, establishing their variational nature.

Findings

Weighted renormalized volume coefficients are variational.

They can be expressed as polynomials of weighted tensors.

The coefficients relate to weighted geometric invariants.

Abstract

We define weighted renormalized volume coefficients and prove that they are variational. We also prove that they can be written as polynomials of weighted extended obstruction tensors, the weighted Schouten tensor, and the weighted Schouten scalar.