On the hyperbolicity of surfaces of general type with small $c_1 ^2$

Surfaces of general type with positive second Segre number $s_2:=c_1^2-c_2>0$ are known by results of Bogomolov to be quasi-hyperbolic i.e. with finitely many rational and elliptic curves. These results were extended by McQuillan in his proof of the Green-Griffiths conjecture for entire curves on such surfaces. In this work, we study hyperbolic properties of minimal surfaces of general type with minimal $c_1^2$, known as Horikawa surfaces. In principle these surfaces should be the most difficult case for the above conjecture as illustrate the quintic surfaces in $\bP^3$. Using orbifold techniques, we exhibit infinitely many irreducible components of the moduli of Horikawa surfaces whose very generic member has no rational curves or even is algebraically hyperbolic. Moreover, we construct explicit examples of algebraically hyperbolic and (quasi-)hyperbolic orbifold Horikawa surfaces.