On the Morse Index with Constraints for Capillary Surfaces

In this paper,we apply the abstract Morse index formulation developed in Tran and Zhou (On the Morse index with constraints: an abstract formulation. Preprint, 2020) to study several optimization set-ups with constraints. In each case, we classify how the general index is related to the index with a constraint. In addition, for capillary surfaces in a Euclidean ball, we obtain an index estimate which recovers stability results of Wang and Xia (Math Ann 374(3–4):1845–1882, 2019) and Gou and Xia (J Geom Anal 31(3):2890–2923, 2021) as special cases. By considering a family of examples, we show that the inequality is sharp. Furthermore, we precisely determine indices with constraints for important examples such as the critical catenoid, round cylinders in a ball, and CMC hypersurfaces with constant scalar curvature in a sphere.