Morse index and heat kernel of constant mean curvature surfaces

1.文特环面的莫尔斯指数我们找到了包含一个连续的平面主曲线族的文特环面的指数的各种上下界。然后，我们证明了一个结果，给出了一个算法，用于计算索引sharpness。双周期极小曲面研究了具有Scherk型端点的嵌入双周期极小曲面的存在性问题。我们将Karcher和Wei的存在性结果推广到更多的情形，并发现了存在性不成立的其它情形.面积极小圆盘的嵌入性我们证明了多边形Jordan洞C满足一定的条件，则以C为边界的最小面积Douglas-Rado圆盘是唯一的，并且是光滑图.利用我们的结果，我们可以将这种方法应用到更广泛的完全悬链末端极小曲面。

1. The Morse Index of Wente ToriWe find various lower and upper bounds for the index of Wente tori that contain a continuous family of planer principal curves. We then prove a result that gives an algorithm for computing the index sharply.2. Doubly Perodic Minimal SurfacesWe consider the question of existence of embedded doubly periodic minimal surfaces with Scherk-type ends. We extend the existance results of Karcher and Wei to more cases and we find other cases where existance dose not hold.3. Embeddedness of area-minimizing disksWe show that a polygonal Jordan cave C satisfies certain conditions, then the least-area Douglas-Rado disk with boundary C is unique and is a smooth graph. With our result we can apply this method to a wider range of complete catenoid-ended minimal surfaces.