Scaling of the elastic energy of small balls for maps between manifolds with different curvature tensors

TL;DR

This paper extends previous results on the scaling of elastic energy for maps between manifolds, showing that the minimal energy scales with the fourth power of thickness and converges to a curvature tensor difference, for general targets.

Contribution

It generalizes prior work to include arbitrary compact Riemannian targets with a curvature difference, answering an open question and extending to noncompact targets under regularity conditions.

Findings

Minimal elastic energy scales as h^4 for small balls.

Convergence to a quadratic form in the curvature difference.

Extension to noncompact targets with regularity conditions.

Abstract

Motivated by experiments and formal asymptotic expansions in the physics literature, Maor and Shachar (J. Elasticity 134 (2019), 149-173) studied the behaviour of a model elastic energy of maps between manifolds with incompatible metrics. For thin objects they analysed the scaling of the minimal elastic energy as a function of the thickness. In particular they showed that for maps from a ball of radius h in an oriented Riemannian manifold to Euclidean space, the infimum of a model elastic energy per unit volume scales like the fourth power of h and after rescaling one gets convergence to a quadratic expression in the curvature tensor R(p), where p denotes the centre of the ball. In this paper we show the same result for general compact oriented Riemannian targets with R(p) replaced by a suitable difference of the curvature tensors in the target and the domain, thus answering Open Question 1 in the paper by Maor and Shachar. The result extends noncompact targets provided they satisfy a uniform regularity condition. A key idea in the proof is to use Lipschitz approximations to define a suitable notion of convergence.