Q-Curvature, Spectral Invariants, and Representation Theory

TL;DR

This paper explores the relationships between Q-curvature, spectral invariants, and representation theory, providing foundational insights into conformal geometry and elliptic operators on manifolds.

Contribution

It offers an introductory, self-contained account connecting functional determinants, Polyakov formulas, and extremal problems in conformal geometry.

Findings

Relation of Q-curvature to spectral invariants

Derivation of Polyakov-type formulas for elliptic operators

Insights into extremal problems in conformal geometry

Abstract

We give an introductory account of functional determinants of elliptic operators on manifolds and Polyakov-type formulas for their infinitesimal and finite conformal variations. We relate this to extremal problems and to the Q-curvature on even-dimensional conformal manifolds. The exposition is self-contained, in the sense of giving references sufficient to allow the reader to work through all details.