A GJMS construction for 2-tensors and the second variation of the total Q-curvature

TL;DR

This paper develops a series of conformally invariant differential operators for 2-tensors and applies them to analyze the second variation of total Q-curvature on certain conformal manifolds, providing explicit formulas for Einstein cases.

Contribution

It introduces new GJMS-type operators for 2-tensors and links them to the second variation of Q-curvature, extending previous conformal geometric methods.

Findings

Constructed conformally invariant operators on 2-tensors.

Described second variation of total Q-curvature for specific manifolds.

Provided explicit formulas for Einstein manifolds.

Abstract

We construct a series of conformally invariant differential operators acting on weighted trace-free symmetric 2-tensors by a method similar to Graham-Jenne-Mason-Sparling's. For compact conformal manifolds of dimension even and greater than or equal to four with vanishing ambient obstruction tensor, one of these operators is used to describe the second variation of the total Q-curvature. An explicit formula for conformally Einstein manifolds is given in terms of the Lichnerowicz Laplacian of an Einstein representative metric.