Volume renormalization for the Blaschke metric on strictly convex domains

TL;DR

This paper studies the volume expansion of the Blaschke metric on strictly convex domains, revealing boundary invariants and conformal structures that are globally invariant.

Contribution

It introduces a new expression for the logarithmic coefficient as an integral of affine invariants and links boundary geometry to conformal invariants.

Findings

L is expressed as an integral of boundary affine invariants

Boundary geometry is characterized as a conformal Codazzi structure

L serves as a global conformal invariant of the boundary

Abstract

We consider the volume expansion of the Blaschke metric, which is a projectively invariant metric on a strictly convex domain in a locally flat projective manifold. When the boundary is even dimensional, we express the logarithmic coefficient L as the integral of affine invariants over the boundary. We also formulate an intrinsic geometry of the boundary as a conformal Codazzi structure and show that L gives a global conformal invariant of the boundary.