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Central Limit Theorem for Intersection Currents of Gaussian Holomorphic Sections
Bin Guo
https://arxiv.org/abs/2603.04588 https://arxiv.org/pdf/2603.04588

On the first eigenvalue of the area Jacobi operator for complex curves in K\"ahler surfaces
Zhenxiao Xie
https://arxiv.org/abs/2602.22744 https://arxiv.org/pdf/2602.22744 https://arxiv.org/html/2602.22744
arXiv:2602.22744v1 Announce Type: new
Abstract: In this paper, we investigate the first eigenvalue $\Lambda_1$ of the area Jacobi operator for complex curves in K\"ahler 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 $\Lambda_1 \geq 2\,\mathfrak{Ric}$, where $\mathfrak{Ric}$ denotes the infimum of the ambient Ricci curvature. For K\"ahler-Einstein surfaces with positive Einstein constant $\mathfrak{c}>0$, this bound reduces to $\Lambda_1 \geq 2\mathfrak{c}$. 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 $g \leq 1$.
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