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The continuity of a surface as a function of its two fundamental forms. (English) Zbl 1042.53003
Summary: 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 \(\mathbb R^2\), then there exists a surface in \(\mathbb R^3\) with these fields as its first and second fundamental forms and this surface is unique up to isometries in \(\mathbb R^3\). We establish here that a surface defined in this fashion varies continuously as a function of its two fundamental forms, for certain natural topologies.

MSC:
53A05 Surfaces in Euclidean and related spaces
58D10 Spaces of embeddings and immersions
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