A new approach to the fundamental theorem of surface theory

The fundamental theorem of surface theory classically asserts that, if a field of positive-definite symmetric matrices (a(alpha beta)) of order two and a field of symmetric matrices (b(alpha beta)) of order two together satisfy the Gauss and Codazzi-Mainardi equations in a simply connected open subset. of R-2, then there exists an immersion theta:omega -> R-3 such that these fields are the first and second fundamental forms of the surface theta(omega), and this surface is unique up to proper isometries in R-3. The main purpose of this paper is to identify new compatibility conditions, expressed again in terms of the functions a(alpha beta) and b(alpha beta), that likewise lead to a similar existence and uniqueness theorem. These conditions take the form of the matrix equation partial derivative(1)A(2) - partial derivative(2)A(1) + A(1)A(2) - A(2)A(1) = 0 in omega, where A(1) and A(2) are antisymmetric matrix fields of order three that are functions of the fields (a(alpha beta)) and (b(alpha beta)), the field (a(alpha beta)) appearing in particular through the square root U of the matrix field C = [GRAPHISC] . 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 del Theta= RU of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension Theta of the unknown immersion 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 [ 22], the unknown immersion theta:omega -> R-3 is found in the present approach in function spaces "with little regularity", such as W-loc(2,p)(omega; 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.