Gradient Flow Selection of CM Points on Symmetric Spaces via Geometric Singular Perturbation Theory

We prove that gradient flows of symmetry-displacement energies on the upper half-plane ℍ = SL2(ℝ)/SO(2) and on the product ℍ × ℍ converge globally to CM points. On ℍ, the involution J: z ↦ −1/z has unique fixed point z = i, and the gradient flow of any weighted combination of distance-to-fixed-point and distance-to-image energies converges to i from every initial condition (Theorem 1). On ℍ × ℍ, the symmetry term creates a one-dimensional normally hyperbolic invariant manifold Fix(J), and a scalar term selects (i, i) within it (Theorem 2). The transverse Hessian of the symmetry penalty is constant along the fixed locus (Proposition 3): Hess⊥ = 4, independent of position, inherited from the sectional curvature of the hyperbolic plane. The selected points correspond to elliptic curves with complex multiplication by the Gaussian integers. All proofs are self-contained, using only Riemannian geometry, Lyapunov theory, and Fenichel’s theorem.