On the first eigenvalue of the area Jacobi operator for complex curves in Kähler surfaces

Abstract: 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 $g \leq 1$.