Let $X$ be a complex minimal surface of general type, *id est* $K_X$ is big and nef. It is well-known that $\displaystyle\int_X3c_2(X)-c_1(X)^2\geq0$, and the equality holds if and only if $X$ is uniformized by $\mathbb{B}\subset\mathbb{C}^2$ (the open ball). Ever in this case: $X$ does not contain neither rational smooth curves nor smooth curves of genus $1$.

In the opposite case, *id est* assuming that $\displaystyle\int_X3c_2(X)-c_1(X)^2>0$, there are examples of such $X$'s which do not contain any smooth rational curves.

**Questions**: let $X$ be a complex minimal surface of general type such that $\displaystyle\int_X3c_2(X)-c_1(X)^2>0$: does $X$ contain any smooth curves of genus $1$? Does the previous answer change if one assumes $\Omega^1_X$ nef but not ample and $K_X$ ample?