On the first eigenvalue of the area Jacobi operator for complex curves in Kähler surfaces
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arXiv
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866918357323743232 |
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| author | Xie, Zhenxiao |
| author_facet | Xie, Zhenxiao |
| contents | In this paper, we investigate the first eigenvalue $Λ_1$ of the area Jacobi operator for complex curves in Kähler surfaces, establishing an extrinsic counterpart to the classical Lichnerowicz theorem for the Laplace-Beltrami operator. By analyzing the second variation of a conformally invariant Willmore-type functional, we derive the lower bound $Λ_1 \geq 2\,\mathfrak{Ric}$, where $\mathfrak{Ric}$ denotes the infimum of the ambient Ricci curvature. For Kähler-Einstein surfaces with positive Einstein constant $\mathfrak{c}>0$, this bound reduces to $Λ_1 \geq 2\mathfrak{c}$. We then explore the equality case, computing the exact dimension of the corresponding first eigenspace in terms of the area, genus, and the dimension of a space of holomorphic sections. This analysis shows that the equality is achieved for all curves of genus $g \leq 1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_22744 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the first eigenvalue of the area Jacobi operator for complex curves in Kähler surfaces Xie, Zhenxiao Differential Geometry 53A10 53C55 58J50 In this paper, we investigate the first eigenvalue $Λ_1$ of the area Jacobi operator for complex curves in Kähler surfaces, establishing an extrinsic counterpart to the classical Lichnerowicz theorem for the Laplace-Beltrami operator. By analyzing the second variation of a conformally invariant Willmore-type functional, we derive the lower bound $Λ_1 \geq 2\,\mathfrak{Ric}$, where $\mathfrak{Ric}$ denotes the infimum of the ambient Ricci curvature. For Kähler-Einstein surfaces with positive Einstein constant $\mathfrak{c}>0$, this bound reduces to $Λ_1 \geq 2\mathfrak{c}$. We then explore the equality case, computing the exact dimension of the corresponding first eigenspace in terms of the area, genus, and the dimension of a space of holomorphic sections. This analysis shows that the equality is achieved for all curves of genus $g \leq 1$. |
| title | On the first eigenvalue of the area Jacobi operator for complex curves in Kähler surfaces |
| topic | Differential Geometry 53A10 53C55 58J50 |
| url | https://arxiv.org/abs/2602.22744 |