AI 中文总结
本文研究闭曲面伪Anosov同胚映射环的纤维同伦类，将Cannon-Thurston边界映射与调和映射及最小面积极小曲面相联系，给出面积曲率界并分析调和能量函数。
AI 中文摘要
设$M_f$为闭双曲$3$--流形，它是闭曲面$S$（亏格至少为2）上伪Anosov同胚$f$的映射环。纤维子群$\pi_1(S)\hookrightarrow \pi_1(M_f)$的包含具有Cannon--Thurston边界映射$\partial\pi_1(S)\cong S^1\to S^2\cong\partial\pi_1(M_f)$，该映射是一条Peano曲线。我们将此映射与纤维同伦类的两个变分代表联系起来：Eells--Sampson和Hartman的调和映射，以及一个最小面积的嵌入极小曲面。同伦映射的等变提升彼此有界距离；因此，对于$S$上的每个标记共形结构，调和代表的提升连续延拓到闭圆盘，其边界值为Cannon--Thurston映射，并且最小面积纤维提升为$\Hh^3$中具有相同边界映射的恰当嵌入极小平面。我们给出最小面积纤维的面积和曲率界，证明其最大主曲率至少为1，并研究调和代表的能量作为Teichmüller空间上的函数、其临界点（即Hopf微分的零点）以及其在单值性下的不变性。
英文摘要
Let $M_f$ be the closed hyperbolic $3$--manifold which is the mapping torus of a pseudo-Anosov homeomorphism $f$ of a closed surface $S$ of genus at least two. The inclusion of the fiber subgroup \(π_1(S)\hookrightarrow π_1(M_f)\) has a Cannon--Thurston boundary map \(\partialπ_1(S)\cong S^1\to S^2\cong\partialπ_1(M_f)\), which is a Peano curve. We relate this map to two variational representatives of the fiber homotopy class: the harmonic map of Eells--Sampson and Hartman, and a least-area embedded minimal surface. Equivariant lifts of homotopic maps are a bounded distance apart; hence, for every marked conformal structure on $S$, the lift of the harmonic representative extends continuously to the closed disk with boundary value the Cannon--Thurston map, and a least-area fiber lifts to a properly embedded minimal plane in $\Hh^3$ with the same boundary map. We give area and curvature bounds for a least-area fiber, show that its largest principal curvature is at least one, and study the energy of the harmonic representatives as a function on Teichmüller space, its critical points, which are the zeros of the Hopf differential, and its invariance under the monodromy.
