We prove that the moduli space of compact genus three Riemann surfaces contains only finitely many algebraically primitive Teichmüller curves. For the stratum \(\Omega\mathcal{M}_{3}(4)\), consisting of holomorphic one-forms with a single zero, our approach to finiteness uses the Harder-Narasimhan filtration of the Hodge bundle over a Teichmüller curve to obtain new information on the locations of the zeros of eigenforms. By passing to the boundary of moduli space, this gives explicit constraints on the cusps of Teichmüller curves in terms of cross-ratios of six points on \(\mathbf{P}^{1}\).

中文翻译：

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Teichmüller在第3类中弯曲，并且可能在\（\ mathbf {G} _ {m} ^ {n} \ times \ mathbf {G} _ {a} ^ {n} \

我们证明了紧属三黎曼曲面的模空间仅包含有限数量的代数本原Teichmüller曲线。对于由全纯单形和单个零组成的\（\ Omega \ mathcal {M} _ {3}（4）\）层，我们的有限性方法是在Teichmüller上使用Hodge束的Harder-Narasimhan滤波曲线以获取有关本征形零点位置的新信息。通过传递到模空间的边界，可以根据\（\ mathbf {P} ^ {1} \）上六个点的交叉比例，对Teichmüller曲线的尖点施加明确的约束。