# Chern-Gauss-Bonnet formula for singular Yamabe metrics in dimension four

## TL;DR

This paper establishes a Chern-Gauss-Bonnet type formula for four-dimensional manifolds with boundary involving the renormalized volume and boundary integrals, revealing conformal invariance properties under certain conditions.

## Contribution

It derives a new geometric formula linking Euler characteristic, renormalized volume, and boundary data for singular Yamabe metrics in four dimensions, extending known results to broader settings.

## Key findings

- The formula involves the Euler characteristic, renormalized volume, and boundary integrals.
- Conformal invariance is shown when the boundary is umbilic.
- Results are extended to asymptotically hyperbolic metrics with constant symmetric functions of the Schouten tensor.

## Abstract

We derive a formula of Chern-Gauss-Bonnet type for the Euler characteristic of a four dimensional manifold-with-boundary in terms of the geometry of the Loewner-Nirenberg singular Yamabe metric in a prescribed conformal class. The formula involves the renormalized volume and a boundary integral. It is shown that if the boundary is umbilic, then the sum of the renormalized volume and the boundary integral is a conformal invariant. Analogous results are proved for asymptotically hyperbolic metrics in dimension four for which the second elementary symmetric function of the eigenvalues of the Schouten tensor is constant. Extensions and generalizations of these results are discussed. Finally, a general result is proved identifying the infinitesimal anomaly of the renormalized volume of an asymptotically hyperbolic metric in terms of its renormalized volume coefficients, and used to outline alternate proofs of the conformal invariance of the renormalized volume plus boundary integral.