Harmonic Gauge on the Space of Riemannian Metrics and Its Role in the Ricci-DeTurck Flow

TL;DR

This paper introduces a harmonic gauge on the space of Riemannian metrics, linking gauge fixing, elliptic operators, and geometric stability, with implications for the Ricci-DeTurck flow.

Contribution

It develops a harmonic gauge framework that clarifies the variational structure and stability properties of the Hilbert-Einstein functional in geometric analysis.

Findings

Harmonic gauge removes divergence terms in the first variation.

Second variation becomes elliptic under the harmonic gauge.

Positivity of the curvature operator implies spectral stability.

Abstract

We develop a harmonic gauge on the space of Riemannian metrics and study its role in the variational and flow-theoretic structure of geometric analysis. We prove that the harmonic gauge eliminates divergence-type terms in the first variation of the Hilbert-Einstein functional and induces a natural elliptic structure for the second variation. As a consequence, positivity of the curvature operator of the second kind implies spectral stability of the functional. This establishes a conceptual link between gauge fixing, elliptic operator theory, and geometric rigidity, and provides a variational counterpart to the Ricci-DeTurck mechanism.