Minimal Hyperbolic Area of Teichmuller Curves in Genus Two

Xiaoyu Su; Yumin Zhong

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arXiv · 2608.01984

Minimal Hyperbolic Area of Teichmuller Curves in Genus Two

Xiaoyu Su·Yumin Zhong

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Abstract

We determine the minimum hyperbolic area of Teichmuller curves arising from holomorphic quadratic differentials on closed Riemann surfaces of genus two. The minimum is 3\pi/5, and it is attained precisely by quadratic differentials q=\omega^2 for which the translation surface (X,\omega) lies in the GL_2^+(R)-orbit of the double-pentagon translation surface. Equivalently, the extremal projective Veech group is the triangle group \Delta(2,5,\infty). The proof combines a small-area classification of noncompact hyperbolic orbifolds with a derivative-preserving affine descent construction for nonsquare quadratic differentials. The three possible nonsquare zero patterns are then excluded by arithmetic, marked-point, and covering obstructions.

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Xiaoyu Su, Yumin Zhong. 2026-08-03. Minimal Hyperbolic Area of Teichmuller Curves in Genus Two. https://arxiv.org/abs/2608.01984

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