On the Morse Index with Constraints II: Applications

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.