In this paper, we establish the sharp local uniform well-posedness of the higher-order nonlinear Schrödinger equations (HNLS) with cubic nonlinear terms \({\rm{i}}{\partial_t}u + {( - {\Delta_g})^m}u = - {\vert u \vert^2}u\) on \({\mathbb{T}^2}\) and \({\mathbb{S}^2}\) . Employing bilinear estimates and lattice point estimates, we prove that the well-posedness thresholds are \({{s}_c}({\mathbb{T}^2}, m) = 0\) and \({{s}_c}({\mathbb{S}^2}, m) = {1 \over 4}\) for any order m ∈ ℕ. In contrast, for Euclidean spaces, it has been shown in [Miao, C., Zhang, B.: Discrete Contin. Dyn. Syst., 17, 181–200 (2006)] that \({{s}_c}({\mathbb{R}^d}, m) = {d \over 2}- m\) if \(m < {d \over 2}\) . These reveal that the geometry of manifolds plays a crucial role in the dynamics of the cubic HNLS.
