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arXiv · 2506.00752

Constructing Two Metrics for Hodge Theory of Constrained Bundles: Harmonic Stability and Magnetic Coercivity

Dongzhe Zheng
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Abstract

We study two metric constructions for Hodge theory on bundles, together with the response of harmonic representatives and magnetic energies to degeneration. Our first result is an $L^1$ coercivity estimate for the covariant derivative on functions of bounded variation: the coercivity constant is positive exactly when the connection has no nonzero parallel section. Extension across internal interfaces then proves the uniform magnetic frustration conjecture of Chakradhar, Gittins, Habib, and Peyerimhoff, and yields positivity of both magnetic Cheeger constants on connected manifolds. For weighted de Rham complexes, we relate the defect in cohomological energy to changes in harmonic representatives. Bracket-generating harmonic vector fields give a quantitative recovery estimate with no lower bound on the varying positive weight. On the Heisenberg nilmanifold, the resulting $H^{-1/2}$ estimate has optimal Sobolev exponent. For circle bundles over surfaces, two connection metrics have identical cohomological Hodge pairings. A curvature upper bound gives a uniform positive Hodge spectral gap even as the adapted metrics lose rank. We identify the scalar spectral limit in every circle character; zero-density regions contribute magnetic Dirichlet-to-Neumann energies that retain their flat holonomy.

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Dongzhe Zheng. 2026-09-19. Constructing Two Metrics for Hodge Theory of Constrained Bundles: Harmonic Stability and Magnetic Coercivity. https://arxiv.org/abs/2506.00752

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