On the fundamental theorem of surface theory under weak regularity assumptions

We consider a symmetric, positive definite matrix field of order two and a symmetric matrix field of order two that together satisfy the Gauss and Codazzi-Mainardi equations in a connected and simply connected open subset of R-2. If these fields are of class C-2 and C-1 respectively, the fundamental theorem of surface theory asserts that there exists a surface immersed in the three-dimensional Euclidean space with the given matrix fields as its first and second fundamental forms. The purpose of this Note is to prove that this theorem still holds true under the weaker regularity assumptions that these fields are of class W-loc(infinity) and L-loc(infinity) respectively, the Gauss and Codazzi-Mainardi equations being then understood in a distributional sense. (C) 2003 Academie des sciences. Published by Elsevier SAS. All rights reserved.