AI 中文总结
针对n≥3时对称空间 $X_n$ 度量 $d_{X_n}$ 缺乏模解释的问题，引入总扩张与希尔伯特-施密特扩张概念，证明保体积平坦环面间利普希茨映射的总扩张可由仿射映射最小化，且对应 $d_{X_n}$。
AI 中文摘要
对称空间 $X_n={\rm SL}(n,\Rb)/{\rm SO}(n)$ 可解释为带标记、单位体积的n维平坦环面的泰希米勒空间，其带有唯一（相差尺度）的 $\rm SL}(n,\Rb)$ 不变度量 $d_{X_n}$。1939年，泰希米勒通过拟共形伸缩的极值映射问题，给出了 $d_{X_2}$（双曲度量）的模解释；但对于 $n\geq3$ 的 $d_{X_n}$，这类模解释仍未解决，因为自然候选者——最小拟共形伸缩、利普希茨常数或总能量——均不适用。本文给出了两种此类模解释，引入了利普希茨映射 $f:M\to N$（在黎曼流形之间）的“总扩张”$\TE(f)\in [0,\infty]$，该概念与格罗莫夫、古思等人发展的“k-伸缩”概念相关。对于n维单位体积平坦环面之间保体积的利普希茨映射 $f:\Tc_0\to\Tc_1$，我们证明：在f的同伦类中，$\TE(f)$ 恰由该类中的仿射映射最小化，且在这些仿射映射上取值为 $d_{X_n}$；对“希尔伯特-施密特扩张”$\HE(f)$（其是M上的简单积分，更具 $L^2$ 特性），我们也证明了类似结果。
英文摘要
The symmetric space $X_n={\rm SL}(n,\Rb)/{\rm SO}(n)$ can be interpreted as the Teichmüller space of marked, unit volume, flat $n$-dimensional tori. It comes with a unique (up to scale) ${\rm SL}(n,\Rb)$-invariant metric $d_{X_n}$. In 1939 Teichmüller gave a modular interpretation of $d_{X_2}$ (the hyperbolic metric) in terms of an extremal mapping problem for quasiconformal dilatation. Such a modular interpretation for $d_{X_n}$ for $n\geq 3$ has remained unaddressed: the natural candidates - minimal quasiconformal dilatation, Lipschitz constant, or total energy - do not work. In this paper we give such a modular interpretation, two in fact. We introduce the {\em total expansion} $\TE(f)\in [0,\infty]$ of a Lipschitz map $f:M\to N$ between Riemannian manifolds, a notion related to the notion of ``$k$-dilatation'' developed by Gromov, Guth and others. For volume-preserving Lipschitz maps $f:\Tc_0\to\Tc_1$ between $n$-dimensional, flat, unit-volume tori, we prove that $\TE(f)$ is minimized in the homotopy class of $f$ precisely by the affine maps in that class and takes on these the value $d_{X_n}$. We prove similar results for the \emph{Hilbert-Schmidt expansion} $\HE(f)$, which is a simple integral over $M$ and has more of an $L^2$ flavor.
Comments15 pages
