A new approach to the fundamental theorem of surface theory

Research output: Journal Publications and ReviewsRGC 21 - Publication in refereed journalpeer-review

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Detail(s)

Original languageEnglish
Pages (from-to)457-473
Journal / PublicationArchive for Rational Mechanics and Analysis
Volume188
Issue number3
Online published8 Dec 2007
Publication statusPublished - Jun 2008

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 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 aαβ 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 (aαβ) and (b αβ ), the field (aαβ ) 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 Wloc2,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.