Algebraic hyperbolicity of surfaces of general type

There is a conjecture that complex surfaces of general type have only finitely many rational and elliptic curves, a property called algebraic hyperbolicity. (This is one sub-case of more general hyperbolicity conjectures of Green-Griffiths and Lang.) Certain cases of this conjecture have been proven, usually utilizing the presence of differential 1-forms (the Bloch theorem) or higher-degree symmetric differentials, as in Bogomolov's result for surfaces satisfying the Chern number inequality $c_1^2>c_2$. We will discuss Bogomolov's technique and give a brief survey of its generalizations by McQuillan and Miyaoka.
Unfortunately, the simplest of the surfaces of general type, projective surfaces in $P^3$ of degree greater than or equal to $5$, do not have any symmetric differentials of any degree. Thus, modified techniques need to be used. We will briefly discuss what is known for these surfaces using the existence of higher-order jet differentials and an approach using reduction to positive characteristic to singular surfaces, which is the subject of ongoing research.