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

In this paper, we investigate the first eigenvalue ${\mathrm{\Lambda }}_1$ 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 ${\mathrm{\Lambda }}_1\ge 2\mathfrak{R}\mathfrak{i}\mathfrak{c}$, where $\mathfrak{R}\mathfrak{i}\mathfrak{c}$ denotes the infimum of the ambient Ricci curvature. For Kähler-Einstein surfaces with positive Einstein constant $\mathfrak{c}>0$, this bound reduces to ${\mathrm{\Lambda }}_1\ge 2\mathfrak{c}$. 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 $g\le 1$.