Morse theory on moduli of curves

Abstract

We provide a new approach to studying the moduli space of curves via Morse theory and hyperbolic geometry, by introducing a family of Morse functions on the moduli space $\overline{\mathcal{M}}_{g,n}$ of stable curves of genus $g$ with $n$ marked point, from the Teichm\"uller theoretic perspective. They are weighted exponential averages of the lengths of all simple closed geodesics. These Morse functions behave well with respect to the Deligne-Mumford stratification of $\overline{\mathcal{M}}_{g,n}$. The critical points can be characterized by a combinatorial property named eutacticity, and the Morse index can be computed accordingly. Also, the Weil-Petersson gradient flow of the Morse functions is well defined on $\overline{\mathcal{M}}_{g,n}$, which can be used to build the Morse theory. These functions might be the first explicit examples of Morse functions on $\overline{\mathcal{M}}_{g,n}$, and the Morse handle decomposition gives rise to the first example of a natural cell decomposition of $\overline{\mathcal{M}}_{g,n}$ in theory, that works for all pairs $(g,n)$.