非欧几里得弹性理论与几何管的近等距嵌入
Non-Euclidean elasticity for rods and almost isometric embeddings of geodesic tubes
AI总结：
本文研究了非欧几里得流形中薄管的弹性能量，通过Γ收敛理论推导了极限能量的表达式，并解决了曲率张量差的最小化问题。
摘要：
我们考虑一个长度为2L的测地线γ在定向黎曼流形(M,g)中，并围绕γ的薄管Ω_h，半径为h。我们研究从Ω_h到另一个定向黎曼流形(M~，g~)的映射u的单位体积弹性能量E_h(u)。该能量E_h基于微分du的平方距离与对应切空间之间保持定向的线性映射集合的差。我们证明了对于序列u_h，当h^{-4}E_h(u_h)保持有界时的紧致性结果，并研究了h→0时h^{-4}E_h(u_h)的Γ极限，这种收敛涉及径向方向的某些放大。该Γ收敛结果扩展了Mora和Müller在欧几里得环境下的薄杆极限能量的工作。我们还得到了极限能量最小值的表达式，作为曲率张量的差的特定二次泛函，从而回答了Maor和Shachar在J.Elasticity 134(2019), pp.149-173中提出的问题。
英文摘要：
We consider a geodesic $γ$ of length $2L$ in an oriented Riemannian manifold $(\mathcal M, g)$ and a thin tube $Ω^*_h$ around $γ$ of radius $h$. We study an 'elastic' energy per unit volume $E_h(u)$ of maps $u$ from $Ω^*_h$ into another oriented Riemannian manifold $(\tilde {\mathcal M},\tilde g)$. The energy $E_h$ is based on the squared distance of the differentials $du$ from the set of orientation preserving linear maps between the corresponding tangent spaces. We prove a compactness result for sequences of maps $u_h$ for which $h^{-4} E_h(u_h)$ remains bounded and we study the $Γ$-Limit of $h^{-4} E_h(u_h)$ as $h \to 0$ with respect to a suitable notion of convergence for $u_h$ that involves certain blow-ups in the radial direction. This $Γ$-convergence result ge\-ne\-ra\-lizes work by Mora and Müller on the limiting energy of thin rods in the Euclidean setting. We also obtain an expression for the minimum of the limiting energy as a specific quadratic functional in the difference of the pullbacks of the curvature tensors of $\mathcal M$ and $\tilde{\mathcal M}$ along the curves $γ$ and $u \circ γ$, respectively, thus answering a question by Maor and Shachar, J. Elasticity 134 (2019), pp. 149--173.
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