Initiated by the work of Uhlenbeck in late 1970s, we study existence, multiplicity and asymptotic behavior for minimal immersions of a closed surface in some hyperbolic three-manifold, with prescribed conformal structure on the surface and second fundamental form of the immersion. We prove several results in these directions, by analyzing the Gauss equation governing the immersion. We determine when existence holds, and obtain unique stable solutions for area minimizing immersions. Furthermore, we find exactly when other (unstable) solutions exist and study how they blow-up. We prove our class of unstable solutions exhibit different blow-up behaviors when the surface is of genus two or greater. We establish similar results for the blow-up behavior of any general family of unstable solutions. This information allows us to consider similar minimal immersion problems when the total extrinsic curvature is also prescribed.

由Uhlenbeck在1970年代后期的工作发起，我们研究了某些双曲三流形中闭合表面的最小浸入的存在，多重性和渐近行为，在表面具有规定的保形结构，并且浸入的第二种基本形式。通过分析控制浸入的高斯方程，我们证明了在这些方向上的一些结果。我们确定何时存在，并获得独特的稳定解决方案以最大程度地减少浸入。此外，我们确切地找到其他（不稳定）解决方案何时存在，并研究它们如何爆炸。我们证明了当表面为两个或更大的类时，我们的不稳定解类别表现出不同的爆炸行为。对于任何不稳定解决方案的一般系列的爆炸行为，我们都建立了相似的结果。