Continuity in Fréchet topologies of a surface as a function of its fundamental forms

A generalization due to Sorin Mardare of the fundamental theorem of surface theory for surfaces with little regularity asserts that, if, for any p > 2, the components of a positive-definite 2 × 2 symmetric matrix field in the space W_loc^1,p and the components of another 2×2 symmetric matrix field in the space L_loc^p satisfy together the Gauss and Codazzi-Mainardi equations in a simply-connected open subset of R², then there exists a surface defined in the three-dimensional Euclidean space E³ by an immersion with components in the space W_loc^2,p, whose first and second fundamental forms are precisely the given matrix fields; besides, this surface is uniquely determined up to isometries in E³.We establish here that a surface defined in this fashion varies continuously as a function of its two fundamental forms for several Fréchet topologies, which include in particular the above spaces W_loc^1,p for the first fundamental form and L_loc^p for the second fundamental form, for any p > 2.