Global analysis of a conformal phase–metric flow with coercive potential

We study a conformal phase–metric flow on a closed surface (\Sigma,g_0), where a scalar phase field \phi and a conformal factor u for the metric g = e^{2u} g_0 evolve by the L^2–gradient flow of a coupled energy with a coercive potential and a geometric coupling. Under a coercivity assumption on V and a smallness condition on the phase–curvature coupling, we prove global existence of classical solutions, uniform H^1–bounds, and convergence of the associated Lyapunov functional along trajectories. Every solution admits a non-empty \omega–limit set consisting of stationary states, and, under additional analyticity and spectral assumptions, we obtain algebraic or exponential convergence to equilibrium via a Łojasiewicz–Simon inequality. For a curvature–coupled model with \mathcal{G}\sim\phi\,R(g), we compute an explicit small-coupling threshold and verify the analytic and spectral hypotheses in a simple geometric setting. The conformal model is presented as a controlled two-dimensional laboratory for phase–metric couplings, rather than as a full Phase–Metric Flow theory. Positioning / status (Dec 2025). This preprint is part of the PDE-controlled technical core of the Phase–Metric Flows (PMF) Project. After this release, the project enters a temporary pause while a journal article is prepared from the analytic components. No algorithmic/computational claim is established here; ICA / Bohr / Meta-Geodesy notes remain programmatic and separate. Navigation (PMF corpus): https://doi.org/10.5281/zenodo.17868144 (this record), https://doi.org/10.5281/zenodo.17816726, https://doi.org/10.5281/zenodo.17352456, https://doi.org/10.5281/zenodo.17505709, https://doi.org/10.5281/zenodo.17641759.