A criterion for being a Teichmüller curve (Q1951839)

Teichmüller curves play an important role in the study of dynamics in polygonal billiards. In this article, the author provides a criterion similar to the original Möller's criterion, to detect whether a complex curve, embedded in the moduli space of Riemann surfaces and endowed with a line subbundle of the Hodge bundle, is a Teichmüller curve, and gives a dynamical proof of this criterion. By the Deligne semisimplicity theorem the Hodge bundle over the curve decomposes into a direct sum of flat subbundles admitting variations of complex polarized Hodge structures of weight 1. Suppose that the restriction of the canonical pseudo-Hermitian form to one of the blocks of the decomposition has rank \((1,r - 1)\). The author establishes an upper bound for the degree of the corresponding holomorphic line bundle in terms of the (orbifold) Euler characteristic of the curve. His criterion claims that if the upper bound is attained, the curve is a Teichmüller curve. For those Teichmüller curves which correspond to strata of abelian differentials that criterion is necessary and sufficient in the sense that if the curve is a Teichmüller curve, then the decomposition of the Hodge bundle necessarily contains a non-trivial block of rank (1,1) corresponding to the tautological line bundle for which the upper bound is attained. The key idea of the criterion is based on \textit{G. Forni}'s [Ann. Math. (2) 155, No. 1, 1--103 (2002; Zbl 1034.37003)] observation that the tautological bundle on a Teichmüller curve is spanned by those vectors of the Hodge bundle which have the maximal variation of the Hodge norm along the Teichmüller flow. He combines this result of Forni with the Bouw-Möller [\textit{I. I. Bouw} and \textit{M. Möller}, Ann. Math. (2) 172, No. 1, 139--185 (2010; Zbl 1203.37049)] version of the Kontsevich formula for the sum of the Lyapunov exponents of the Hodge bundle along the Teichmüller geodesic flow. Theorem. If \(\chi(C) \geq 0,\) then \(C\) is not a Teichmüller curve. Suppose that \(\chi(C) < 0\). For any flat subbundle \(L\) of the Hodge bundle over \(C\) satisfying the above assumptions, one has \(\deg \overline{L^{1,0}} \leq -\frac{\chi(C)}{2}\). If the equality is attained, then \(C\) is a Teichmüller curve, and the line bundle \(L^{1,0},\) is the tautological bundle. Any Teichmüller curve corresponding to a stratum of abelian differentials admits a flat subbundle \(L\) of the Hodge bundle satisfying the above conditions, such that \(\deg\overline{L^{1,0}} = -\frac{\chi(C)}{2}.\)