Willmore surfaces in 4-dimensional conformal manifolds
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2023
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| _version_ | 1866910799834906624 |
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| author | Wang, Changping Xie, Zhenxiao |
| author_facet | Wang, Changping Xie, Zhenxiao |
| contents | This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_00846 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Willmore surfaces in 4-dimensional conformal manifolds Wang, Changping Xie, Zhenxiao Differential Geometry 53C18, 53C42, 49Q10 This paper is dedicated to the exploration of the conformal Willmore functional for surfaces within 4-dimensional conformal manifolds. We provide a detailed calculation of both the first and second variations, and present the Euler-Lagrange equation of this functional in a conformally invariant form. Utilizing the second variation formula we derived, we demonstrate that the Clifford torus in $\mathbb{C}P^2$ is strictly Willmore-stable. This finding strongly supports the conjecture proposed by Montiel and Urbano [J. reine angew. Math. 546 2002, 139-154], which posits that the Clifford torus in $\mathbb{C}P^2$ minimizes the Willmore functional among all tori. Moreover, by applying our formula to complex curves in $\mathbb{C}P^2$, we establish that the first nonzero eigenvalue of the Jacobi operator is at least 12. In the context of 4-dimensional locally symmetric spaces, we construct several holomorphic differentials to show that among all minimal 2-spheres, only those super-minimal ones can be Willmore. |
| title | Willmore surfaces in 4-dimensional conformal manifolds |
| topic | Differential Geometry 53C18, 53C42, 49Q10 |
| url | https://arxiv.org/abs/2306.00846 |