GJMS operators and Q-curvature for conformal Codazzi structures

TL;DR

This paper develops GJMS operators and Q-curvature for conformal Codazzi structures on convex hypersurfaces, linking total Q-curvature to volume expansion and deriving variation formulas for domain deformations.

Contribution

It introduces the construction of GJMS operators and Q-curvature for conformal Codazzi structures using ambient metrics, a novel approach in this geometric setting.

Findings

Total Q-curvature relates to volume expansion of the Blaschke metric.

Derived first and second variation formulas for convex domain deformations.

Established connections between Q-curvature and geometric invariants.

Abstract

The conformal Codazzi structure is an intrinsic geometric structure on strictly convex hypersufaces in a locally flat projective manifold. We construct the GJMS operators and the Q-curvature for conformal Codazzi structures by using the ambient metric. We relate the total Q-curvature to the logarithmic coefficient in the volume expansion of the Blaschke metric, and derive the first and second variation formulas for a deformation of strictly convex domains.