On O(p)×O(q)-invariant constant mean curvature hypersurfaces with singularity

The paper proves Hsiang’s conjecture using a cylindrical blow-up and invariant manifold analysis of a 3D vector field derived from the O(p)×O(q)-invariant CMC system, yielding uniqueness of the global type-E solution, its tangent α0 with tan α0=√((p−1)/(q−1)), and the origin curvatures ±2H(p+q−1)/(3p+3q−4), as well as the minimal cone case H=0 with stated principal curvatures. These statements and constants agree with the model’s conclusions. The model presents a different, weighted-curvature/Frobenius approach that reproduces the same leading-order asymptotics and uniqueness of the cusp germ, and gives the same curvature constants and minimal-cone data. However, the model’s local expansion omits the r^2 term (which the paper computes and is generally nonzero unless p=q) and its global extendability argument is more heuristic than the paper’s dynamical-systems proof. Overall, the paper’s argument is correct and complete, and the model’s solution is essentially correct on the main claims but would benefit from minor corrections and tighter justification. Key paper facts used: the ODE (1.1) and angle system (2.6)–(2.7), blow-up vector field (2.8), the determination of α0, and curvature computation at the origin (Proposition 2.6), and the minimal case (Theorem 1.2) .