On the Morse Index with Constraints II: Applications
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arXiv
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| Format: | Preprint |
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2020
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| _version_ | 1866915744738967552 |
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| author | Tran, Hung Zhou, Detang |
| author_facet | Tran, Hung Zhou, Detang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_05951 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | On the Morse Index with Constraints II: Applications Tran, Hung Zhou, Detang Differential Geometry Analysis of PDEs Optimization and Control 53A10, 53C42, 49Q10, 58E05, 76B45 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. |
| title | On the Morse Index with Constraints II: Applications |
| topic | Differential Geometry Analysis of PDEs Optimization and Control 53A10, 53C42, 49Q10, 58E05, 76B45 |
| url | https://arxiv.org/abs/2010.05951 |