RECOVERY OF A SURFACE WITH BOUNDARY AND ITS CONTINUITY AS A FUNCTION OF ITS TWO FUNDAMENTAL FORMS

Abstract

If a field A of class [Formula: see text] of positive-definite symmetric matrices of order two and a field B of class [Formula: see text] of symmetric matrices of order two satisfy together the Gauss and Codazzi–Mainardi equations in a connected and simply-connected open subset ω of ℝ2, then there exists an immersion [Formula: see text], uniquely determined up to proper isometries in ℝ3, such that A and B are the first and second fundamental forms of the surface θ(ω). Let [Formula: see text] denote the equivalence class of θ modulo proper isometries in ℝ3 and let [Formula: see text] denote the mapping determined in this fashion. The first objective of this paper is to show that, if ω satisfies a certain "geodesic property" (in effect a mild regularity assumption on the boundary of ω) and if the fields A and B and their partial derivatives of order ≤ 2 (respectively, ≤ 1), have continuous extensions to [Formula: see text], the extension of the field A remaining positive-definite on [Formula: see text], then the immersion θ and its partial derivatives of order ≤ 3 also have continuous extensions to [Formula: see text]. The second objective is to show that, if ω satisfies the geodesic property and is bounded, the mapping ℱ can be extended to a mapping that is locally Lipschitz-continuous with respect to the topologies of the Banach spaces [Formula: see text] for the continuous extensions of the matrix fields (A, B), and [Formula: see text] for the continuous extensions of the immersions θ.