On general type surfaces with $$$$ and $$c_2 = 3 p_g$$c2=3pg

Let S be a minimal surface of general type with irregularity \(q(S) = 1\). Well-known inequalities between characteristic numbers imply that $$\begin{aligned} 3 p_g(S) \le c_2(S) \le 10 p_g(S), \end{aligned}$$ where \(p_g(S)\) is the geometric genus and \(c_2(S)\) the topological Euler characteristic. Surfaces achieving equality for the upper bound are classified, starting with work of Debarre. We study equality in the lower bound, showing that for each \(n \ge 1\) there exists a surface with \(q = 1\), \(p_g = n\), and \(c_2 = 3n\). The moduli space \(\mathcal {M}_n\) of such surfaces is a finite set of points, and we prove that \(\#\mathcal {M}_n \rightarrow \infty \) as \(n \rightarrow \infty \). Equivalently, this paper studies the number of closed complex hyperbolic 2-manifolds of first betti number 2 as a function of volume; in particular, such a manifold exists for every possible volume.