论文标题
在带有约束的摩尔斯指数上II：应用程序
On the Morse Index with Constraints II: Applications
论文作者
论文摘要
这是我们序列中的第二篇论文。在这里，我们将上一篇文章开发的抽象摩尔斯索引索引公式用于研究具有约束的几种优化设置，包括I型或/和/和/和II型考虑。一个共同的主题是，临界点属于毛细血管表面的家族，该家族由恒定平均曲率定义，并以固定角度与周围歧管相交。在每种情况下，我们都会对一般索引与索引与索引的关系进行分类。对于欧几里得球中的毛细血管表面，我们获得了索引估计值，该索引估计恢复了G. Wang和C. Xia和J. Gou和C. Xia的稳定性结果。通过考虑一个例子，我们表明不平等也很敏锐。此外，我们精确地确定了重要示例的约束索引，例如临界catenoid，球中的圆缸以及球体中持续曲率的CMC表面。
This is the second paper in our sequence. Here, we apply our abstract Morse index formulation developed in the previous paper to study several optimization set-ups with constraints, including type I or/and type II considerations. A common theme is that critical points belong to the family of capillary surfaces, defined by constant mean curvature and intersecting the ambient manifold at a fixed angle. In each case, we classify how the general index is related to the index with a constraint. For capillary surfaces in a Euclidean ball, we obtain an index estimate which recovers stability results of G. Wang and C. Xia and J. Gou and C. Xia as special cases. By considering a family of examples, we show that inequality is also sharp. Furthermore, we precisely determine indices with constraints for important examples such as the critical catenoid, round cylinders in a ball, and CMC surfaces with constant curvature in a sphere.
