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.
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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.