Gradient Flow Selection of the Gaussian CM Point on H and H × H

This paper studies gradient flows of energy functionals on the hyperbolic plane H = SL(2,R)/SO(2) and on H × H, built from two geometrically natural terms: a scalar energy measuring distance to a base point, and a symmetry energy measuring the distance from a point to its image under the isometry σ(z) = −1/z. The main results show that for all positive weights the functional is strictly geodesically convex, has a unique critical point, and its gradient flow converges globally — from every initial condition — to the point i (respectively (i,i)), which is the complex-multiplication point for the Gaussian integers Z[i]. A closed form for the symmetry penalty, |z²+1|²/(2 Im(z)²), makes positivity and properness immediate. The paper also records the quantitative two-timescale structure: the transverse Hessian along the fixed locus is constant at 4 = 4|K|, and the linearization has tangential eigenvalues −α and transverse eigenvalues −(α+4β). Proofs use only Riemannian geometry, convexity, and Lyapunov theory. Version 2 corrects the rate constants of the earlier version and closes a gap in the convergence argument; changes are documented in §8.