RECOVERY OF A SURFACE WITH BOUNDARY AND ITS CONTINUITY AS A FUNCTION OF ITS TWO FUNDAMENTAL FORMS
Analysis and Applications
If a field A of class C 2 of positive-definite symmetric matrices of order two and a field B of class C 1 of symmetric matrices of order two satisfy together the Gauss and Codazzi-Mainardi equations in a connected and simply-connected open subset ω of R 2 , then there exists an immersion θ ∈ C 3 (ω; R 3 ), uniquely determined up to proper isometries in R 3 , such that A and B are the first and second fundamental forms of the surface θ(ω). Letθ denote the equivalence class of θ modulo proper
... θ modulo proper isometries in R 3 and let F : (A, B) →θ 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, resp. ≤ 1, have continuous extensions to ω, the extension of the field A remaining positive-definite on ω, then the immersion θ and its partial derivatives of order ≤ 3 also have continuous extensions to ω. 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 C 2 (ω) × C 1 (ω) for the continuous extensions of the matrix fields (A, B), and C 3 (ω) for the continuous extensions of the immersions θ.