The Fundamental Theorem of Surface Theory for Surfaces with Little Regularity

Consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply connected open subset of R ². If the matrix fields are respectively of class C ² and C ¹, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with these fields as its first and second fundamental forms. The purpose of this paper is to prove that this theorem still holds under the weaker regularity assumptions that the matrix fields are respectively of class W ^1,∞ _loc and L ^∞ _loc, the Gauss and Codazzi-Mainardi equations being then understood in a distributional sense.