AI 中文总结
该研究确定亏格2的Teichmüller曲线的极小双曲面积为3π/5，明确其取得条件，通过结合轨形分类与仿射下降构造，排除非平方零点模式完成证明。
AI 中文摘要
我们确定了由亏格2闭黎曼曲面上全纯二次微分导出的Teichmüller曲线的极小双曲面积，该最小值为3π/5，仅当二次微分q=ω²时取得，对应的平移曲面(X,ω)属于双五边形平移曲面的GL₂⁺(ℝ)轨道；等价地，极值射影Veech群为三角群Δ(2,5,∞)。证明结合了非紧双曲轨形的小面积分类，以及非平方二次微分的保导数仿射下降构造，随后通过算术、标记点和覆盖障碍排除了三种可能的非平方零点模式。
英文摘要
We determine the minimum hyperbolic area of Teichmuller curves arising from holomorphic quadratic differentials on closed Riemann surfaces of genus two. The minimum is 3π/5, and it is attained precisely by quadratic differentials q=ω^2 for which the translation surface (X,ω) lies in the GL_2^+(R)-orbit of the double-pentagon translation surface. Equivalently, the extremal projective Veech group is the triangle group Δ(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.
Comments34 pages. Comments welcome
