Continuity in Fréchet topologies of a surface as a function of its fundamental forms

Research output: Journal Publications and Reviews (RGC: 21, 22, 62)21_Publication in refereed journalpeer-review

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Original languageEnglish
Pages (from-to)243-265
Number of pages23
Journal / PublicationJournal des Mathematiques Pures et Appliquees
Online published6 Dec 2019
Publication statusPublished - Oct 2020


A generalization due to Sorin Mardare of the fundamental theorem of surface theory for surfaces with little regularity asserts that, if, for any p > 2, the components of a positive-definite 2 × 2 symmetric matrix field in the space Wloc1,p and the components of another 2×2 symmetric matrix field in the space Llocp satisfy together the Gauss and Codazzi-Mainardi equations in a simply-connected open subset of R2, then there exists a surface defined in the three-dimensional Euclidean space E3 by an immersion with components in the space Wloc2,p, whose first and second fundamental forms are precisely the given matrix fields; besides, this surface is uniquely determined up to isometries in E3.We establish here that a surface defined in this fashion varies continuously as a function of its two fundamental forms for several Fréchet topologies, which include in particular the above spaces Wloc1,p for the first fundamental form and Llocp for the second fundamental form, for any p > 2.

Research Area(s)

  • Differential geometry of surfaces, Fundamental forms, Nonlinear Korn inequalities