The Fundamental Theorem of Surface Theory for Surfaces with Little Regularity

Abstract

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 R2. If the matrix fields are respectively of class C2 and C1, 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 W1,∞loc and L∞loc, the Gauss and Codazzi-Mainardi equations being then understood in a distributional sense.