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 Ξ=q(z) dz3 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,Ξ) 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.
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BibTeX
@misc{shi2026polynomial,
title = {{Polynomial area growth for special Kähler metrics}},
author = {Yiqian Shi and Ke Wang and Bin Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32087},
eprint = {2609.32087},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32087},
}
RIS
TY - UNPB
TI - Polynomial area growth for special Kähler metrics
AU - Yiqian Shi
AU - Ke Wang
AU - Bin Xu
PY - 2026
DA - 2026/09/25
PB - arXiv
DO - 10.48550/arxiv.2609.32087
UR - https://arxiv.org/abs/2609.32087
AB - Let g=w|dz|^2 be an affine special Kähler metric on a punctured disc and let Ξ=q(z) dz^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,Ξ) 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.
ER -