  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.
