Abstract
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.
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Notes
It is also called weak Morse index in literature.
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Acknowledgements
H. T. was partially supported by a Simons Foundation Grant [709791] and National Science Foundation Grant [DMS-2104988]. He also would like to thank Richard Schoen for valuable discussion on the topic and the Vietnam Institute for Advanced Study in Mathematics for its support and hospitality. D. Z. was partially supported by Faperj and CNPq of Brazil.
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Tran, H., Zhou, D. On the Morse Index with Constraints for Capillary Surfaces. J Geom Anal 33, 110 (2023). https://doi.org/10.1007/s12220-022-01135-3
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DOI: https://doi.org/10.1007/s12220-022-01135-3