Polynomial area growth for special Kähler metrics

Abstract

Let $g=w|dz{|}^2$ be an affine special Kähler metric on a punctured disc and let $\Xi =q(z)d{z}^3$ be its associated holomorphic cubic differential. We prove that if the $g$-area of concentric punctured neighbourhoods grows at most polynomially, then $q$ is meromorphic at the puncture. More precisely, an area bound of order $\eps^{-N}$ forces every pole of $q$ to have order strictly less than $N+3$. Hence an essential singularity of the cubic differential rules out every polynomial area upper bound. The proof associates to $(g,\Xi )$ a curvature $-1$ conformal pseudometric and applies the Ahlfors--Schwarz lemma, followed by the submean inequality for holomorphic functions. We also discuss the identically zero cubic differential and the borderline logarithmic growth responsible for the strict pole-order bound.