A New Approach to the Fundamental Theorem of Surface Theory

Abstract

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a αβ ) of order two and a field of symmetric matrices (b αβ ) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset ω of $${\mathbb{R}}^{2}$$ , then there exists an immersion $${\bf \theta}:\omega \to {\mathbb{R}}^{3}$$ such that these fields are the first and second fundamental forms of the surface $${\bf \theta}(\omega)$$ , and this surface is unique up to proper isometries in $${\mathbb{R}}^3$$ . The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a αβ and b αβ , that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation $$\partial{\bf A}_2-\partial_2{\bf A}_1+{\bf A}_1{\bf A}_2-{\bf A}_2{\bf A}_1={\bf 0}\,{\rm in}\,\omega,$$ where A 1 and A 2 are antisymmetric matrix fields of order three that are functions of the fields (a αβ ) and (b αβ ), the field (a αβ ) appearing in particular through the square root U of the matrix field $${\bf C} = \left(\begin{array}{lll} a_{11} & a_{12} & 0\\ a_{21} & a_{22} & 0\\ 0 & 0 & 1\end{array}\right).$$ The main novelty in the proof of existence then lies in an explicit use of the rotation field R that appears in the polar factorization $${\bf \nabla}{\bf \Theta}={\bf RU}$$ of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension $${\bf \Theta}$$ of the unknown immersion $${\bf \theta}$$ . In this sense, the present approach is more “geometrical” than the classical one. As in the recent extension of the fundamental theorem of surface theory set out by S. Mardare [20–22], the unknown immersion $${\bf \theta}: \omega \to {\mathbb{R}}^3$$ is found in the present approach to exist in function spaces “with little regularity”, such as $$W^{2,p}_{\rm loc}(\omega;{\mathbb{R}}^3)$$ , p > 2. This work also constitutes a first step towards the mathematical justification of models for nonlinearly elastic shells where rotation fields are introduced as bona fide unknowns.