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Shimura-Teichmüller Curves in Genus 5

We prove that there are no Shimura-Teichmüller curves generated by genus five translation surfaces, thereby completing the classification of Shimura-Teichmüller curves in general. This was conjectured by Möller in his original work introducing Shimura-Teichmüller curves. Moreover, the property of being a Shimura-Teichmüller curve is equivalent to having completely degenerate Kontsevich-Zorich spectrum. The main new ingredient comes from the work of Hu and the second named author, which facilitates calculations of higher order terms in the period matrix with respect to plumbing coordinates. A large computer search is implemented to exclude the remaining cases, which must be performed in a very specific way to be computationally feasible.