以最小变形嵌入小区域内的曲面
Embedding surfaces inside small domains with minimal distortion
摘要：
给定二维黎曼流形$\mathcal{M},\mathcal{N}$，对于一类畸变泛函，我们证明了嵌入$\mathcal{M} \to \mathcal{N}$的畸变的一个下界，它是由面积差$V_{\mathcal{N}}/V_{\mathcal{M}}$表示的.对于$V_{\mathcal{N}}/V_{\mathcal{M}} \ge 1/4$，如果存在相似性，则相似性是达到上界的唯一能量最小化映射，而对于$V_{\mathcal{N}}/V_{\mathcal{M}} \le 1/4$，存在非相似最小化器。我们刻画了达到上界的映射，并构造了圆盘间的显式非位似极小元。然后，我们证明这两个制度的稳定性结果。最后，我们分析了其他家庭的扭曲泛函。特别是，我们的特点是一个家庭的泛函中没有相变的极小发生; homotheties是能量最小的所有值的$V_{\mathcal{N}}/V_{\mathcal{M}}$，只要它们存在。
英文摘要：
Given two-dimensional Riemannian manifolds $\mathcal{M},\mathcal{N}$, we prove a lower bound on the distortion of embeddings $\mathcal{M} \to \mathcal{N}$, in terms of the areas' discrepancy $V_{\mathcal{N}}/V_{\mathcal{M}}$, for a certain class of distortion functionals. For $V_{\mathcal{N}}/V_{\mathcal{M}} \ge 1/4$, homotheties, provided they exist, are the unique energy minimizing maps attaining the bound, while for $V_{\mathcal{N}}/V_{\mathcal{M}} \le 1/4$, there are non-homothetic minimizers. We characterize the maps attaining the bound, and construct explicit non-homothetic minimizers between disks. We then prove stability results for the two regimes. We end by analyzing other families of distortion functionals. In particular we characterize a family of functionals where no phase transition in the minimizers occurs; homotheties are the energy minimizers for all values of $V_{\mathcal{N}}/V_{\mathcal{M}}$, provided they exist.
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