Abstract
We solve several problems that involve imposing metrics on surfaces. The problem of a strip with a linear metric gradient is formulated in terms of a Lagrangian similar to those used for spin systems. We are able to show that the low energy state of long strips is a twisted helical state like a telephone cord. We then extend the techniques used in this solution to two–dimensional sheets with more general metrics. We find evolution equations and show that when they are not singular, a surface is determined by knowledge of its metric, and the shape of the surface along one line. Finally, we provide numerical evidence by minimizing a suitable energy functional that once these evolution equations become singular, either the surface is not differentiable, or else the metric deviates from the target metric.
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Marder, M., Papanicolaou, N. Geometry and Elasticity of Strips and Flowers. J Stat Phys 125, 1065–1092 (2006). https://doi.org/10.1007/s10955-006-9087-x
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DOI: https://doi.org/10.1007/s10955-006-9087-x