On Harnack's theorem and extensions: A geometric proof and applications

Harnack's theorem states that the fixed points of an orientation reversing involution of a compact orientable surface of genus $g$ are a set of $k$ disjoint simple closed geodesic where $0\leq k\leq g+1$. The first goal of this article is to give a purely geometric, complete and self-contained proof of this fact. In the case where the fixed curves of the involution do not separate the surface, we prove an extension of this theorem, by exhibiting the existence of auxiliary invariant curves with interesting properties. Although this type of extension is well known (see, for instance, Comment. Math. Helv. 57(4): 603-626 (1982) and Transl. Math. Monogr., vol. 225, Amer. Math. Soc., Providence, RI, 2004), our method also extends the theorem in the case where the surface has boundary. As a byproduct, we obtain a geometric method on how to obtain these auxiliary curves. As a consequence of these constructions, we obtain results concerning presentations of Non-Euclidean crystallographic groups and a new proof of a result on the set of points corresponding to real algebraic curves in the compactification of the Moduli space of complex curves of genus $g$, $\overline{\mathcal{M}_{g}}$. More concretely, we establish that given two real curves there is a path in $\overline{\mathcal{M}_{g}}$ which passes through at most two singular curves, a result of M. Seppälä (Ann. Sci. École Norm. Sup. (4), 24(5), 519-544 (1991)).