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 _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 ℝ², then there exists an immersion θ: ω → ℝ³ such that these fields are the first and second fundamental forms of the surface θ(ω), and this surface is unique up to proper isometries in ℝ³. 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 ∂₁A₂ - ∂₂A₁ + A ₁A₂ - A₂A₁ = 0 in ω, where A₁ and A₂ 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 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 ∇Θ = RU of the restriction to the unknown surface of the gradient of the canonical three-dimensional extension Θ of the unknown immersion θ: 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 θ: ω → ℝ³ is found in the present approach to exist in function spaces "with little regularity", such as W^loc2,p (ω; Rdbl;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.