不变包络与几何变分原理
Invariant Hulls and Geometric Variational Principles
AI总结：
本文通过内变分构造任意泛函的不变包络，研究其性质，并在高维情形下对体积泛函得出适度结论，同时提出若干问题与猜想。
摘要：
我们通过参数化来研究定义在流形上的泛函。若这些泛函从几何观点来看是有意义的，它们理应在重新参数化下保持不变。具有这种不变性的标准局部积分泛函是众所周知的。然而，我们关注的是从给定的任意泛函到其不变实现或不变包络的过渡，这通过使用内变分来实现，其方式与向量变分学中被积函数的凸包络或拟凸包络类似。然而，这两个过程在本质上是截然不同的。在考察了泛函与其不变实现之间相互关系的一些基本、有趣的一般性质之后，我们处理一维情形以初步熟悉这种变换和计算，然后再进入高维情形。正如人们所预料的，在高维情形中的显式计算要困难得多，甚至可能无法进行，因为需要处理向量变分问题。特别地，我们能够在一般的$N$维情形下，尤其是二维情形$N=2$时，对流形一部分的体积泛函得出一些适度的结论。文中还陈述了各种问题和猜想。
英文摘要：
We investigate functionals defined on manifolds through parameterizations. If they are to be meaningful, from a geometrical viewpoint, they ought to be invariant under reparameterizations. Standard, local, integral functionals with this invariance property are well-known. We would like to focus though on the passage from a given arbitrary functional to its invariant realization or invariant hull through the use of inner-variations, much in the same way as with the convex or quasiconvex hulls of integrands in the vector Calculus of Variations. These two processes are, however, very different in nature. After examining some basic, interesting, general properties about the mutual relationship between a functional and its invariant realization, we deal with the one dimensional case to gain some initial familiarity with such a transformation and calculations, before proceeding to the higher dimensional situation. As one would anticipate, explicit computations in the latter are much harder to perform, if not impossible, as one is to work with vector variational problems. In particular, we are able to reach some modest conclusion about the volume functional of a piece of a manifold in the general $N$-dimensional situation, especially in the two-dimensional case $N=2$. Various problems and conjectures are stated along the way.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
