Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space

Raz Kupferman; Yossi Shamai

doi:10.1007/s11856-011-0187-1

Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space

Raz Kupferman^*, Yossi Shamai

^*Corresponding author for this work

Einstein Institute of Mathematics

Research output: Contribution to journal › Article › peer-review

5 Scopus citations

Abstract

We study a geometric problem that originates from theories of nonlinear elasticity: given a non-flat n-dimensional Riemannian manifold with boundary, homeomorphic to a bounded subset of ℝⁿ, what is the minimum amount of deformation required in order to immerse it in a Euclidean space of the same dimension? The amount of deformation, which in the physical context is an elastic energy, is quantified by an average over a local metric discrepancy. We derive an explicit lower bound for this energy for the case where the scalar curvature of the manifold is non-negative. For n = 2 we generalize the result for surfaces of arbitrary curvature.

Original language	English
Pages (from-to)	135-156
Number of pages	22
Journal	Israel Journal of Mathematics
Volume	190
Issue number	1
DOIs	https://doi.org/10.1007/s11856-011-0187-1
State	Published - Aug 2012

Bibliographical note

Funding Information:
Acknowledgements. We are very much indebted to E. Farjoun, O. H Hald and J. Solomon for their continual assistance. We have also benefited from discussions with L. C. Evans, D. Kazhdan and P. Li. This research was partially supported by the Israel Science Foundation.

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Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space

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