Recovery of immersions from their metric tensors and nonlinear Korn inequalities: A brief survey

Abstract

The authors discuss the existence and uniqueness up to isometries of E ⁿ of immersions ϕ: Ω ⊂ R ⁿ → E ⁿ with prescribed metric tensor field (g _ij): Ω → S_> ⁿ, and discuss the continuity of the mapping (g _ij) → ϕ defined in this fashion with respect to various topologies. In particular, the case where the function spaces have little regularity is considered. How, in some cases, the continuity of the mapping (g _ij) → ϕ can be obtained by means of nonlinear Korn inequalities is shown.

Keywords

Isometric immersions / Nonlinear Korn inequalities / Metric tensor

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