Incompatible elasticity and the immersion of non-flat Riemannian manifolds in Euclidean space |
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Authors: | Raz Kupferman Yossi Shamai |
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Institution: | 1.Institute of Mathematics,The Hebrew University of Jerusalem,Jerusalem,Israel;2.Mathematics Institute,University of Warwick,Coventry,UK |
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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 ? n , 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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