The Continuity of a surface as a function of its two fundamental forms

The fundamental theorem of surface theory asserts that, if a field of positive definite symmetric matrices of order two and a field of symmetric matrices of order two together satisfy the Gauß and Codazzi-Mainardi equations in a connected and simply connected open subset of R², then there exists a surface in R³ with these fields as its first and second fundamental forms and this surface is unique up to isometries in R³. We establish here that a surface defined in this fashion varies continuously as a function of its two fundamental forms, for certain natural topologies. © 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.