The isometric immersion problem for two-dimensional Riemannian manifolds in the three-dimensional Euclidean space in differential geometry can be formulated as solving the so-called Gauss-Codazzi system. The system is of mixed hyperbolic-elliptic type: hyperbolic for negative Gauss curvature and elliptic for positive one. A local \(C^2\) solution of the Gauss-Codazzi system on the region with negative Gauss curvature was constructed in the previous work (Y. Hu, H. Guo, and X. Qin, Calc. Var. 64 (2025) 142). In this paper, we study the uniform regularity of the solution up to the zero Gauss curvature curve. The main difficulty is the need to establish the precise estimates of the hidden singularity in the governing equations caused by the degeneracy of Gauss curvature. Based on the fluid dynamic framework, we apply the characteristic decomposition technique and the bootstrap idea to derive the uniform regularity of some “bad" terms in a partial hodograph plane. It is analyzed the loss of regularity of the solution returning to the physical plane due to the singularity of the coordinate transformation. We show that the solution is uniformly \(C^{1,\mu }\) up to the zero Gauss curvature curve for \(\mu \in (0,1/3)\) .
