On the first eigenvalue of the area Jacobi operator for complex curves in K\"ahler surfaces

TL;DR

This paper establishes a lower bound for the first eigenvalue of the area Jacobi operator for complex curves in K"ahler surfaces, linking it to ambient Ricci curvature, and characterizes cases of equality, especially for genus 0 or 1.

Contribution

It provides an extrinsic eigenvalue estimate for complex curves in K"ahler surfaces and characterizes the equality case in terms of geometric and topological properties.

Findings

Lower bound $ ext{Lambda}_1 \, \geq \, 2\mathfrak{Ric}$ established.

Equality case analyzed for genus $g \leq 1$.

Explicit dimension of eigenspace computed in terms of geometry.

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 $g \leq 1$.