Let $S=S_{g}$ be a surface of genus $g>1$, and $\Teich(S)$ be the Teichm\"uller space endowed with the Weil-Petersson metric and $\Mod(S)$ be the mapping class group of $S$. This dissertation mainly consists of three parts. <br/> <br/> <br/> The first part is to study the translation lengths of parabolic isometries on complete proper visible CAT(0) spaces and their applications. We show that the translation length of parabolic isometry is always zero in the visible case. The first application is that the mapping class groups $\Mod(S_{g})$ of $S_{g}$ $(g\geq 3)$ which properly discontinuously act on a complete proper visible CAT(0) space by isometries have zero translation length (every element in $\Mod(S_{g})$ has zero translation length). We also apply the zero property to giving a criterion for closed two-dimensional manifolds with bounded geometry. The third application is to give a negative answer to P. Eberlein's conjecture which says that \textsl{a complete open manifold $M$ with sectional curvature $-1\leq K_{M}\leq 0$ and finite volume is visible if the universal covering space $\tilde{M}$ of $M$ contains no imbedded flat half planes}.<br/> <br/> <br/> Secondly, we show that, fix $X,Y\in \Teich(S)$, for any $\phi \in \Mod(S)$, there exists a positive integer $k$ depending on $\phi$ such that the sequence of the directions of geodesics connecting $X$ and $\phi^{kn}\circ{Y}$ is convergent in the visual sphere of $X$. In particular the ``limit'' of the sequence of geodeiscs joining $X$ and $\phi^{kn}\circ{Y}$ exists in some sense, a geometric description for the limit is provided in this dissertation.<br/> <br/> <br/> The third part is to show that the Riemannian sectional curvature operator of $\Teich(S)$ is non-positive definite. As an application we show that any twist harmonic map with respect to $\Mod(S)$ from rank-one hyperbolic spaces $H_{Q,m}=Sp(m,1)/Sp(m)\cdot Sp(1)$ or $H_{O,2}=F_{4}^{-20}/SO(9)$ into $\Teich(S)$ must be a constant map.
Wu, Yunhui,
"Isometries on CAT(0) spaces, iteration of mapping classes and Weil-Petersson geometry"
(2012).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0QJ7FM6
