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

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.