On the Brody lemma (Q938285)

Brody's characterization of compact Kobayashi hyperbolic manifolds says that a compact complex manifold \(X\) is Kobayashi hyperbolic if and only if it does not contain non-constant entire curves. This follows from Brody's lemma: if \(f_n:\Delta\to X\) is a sequence of holomorphic maps from the unit disk \(\Delta\subset{\mathbb C}\) into \(X\) such that \(\|f'_n(0)\|\to+\infty\) (where the norm is computed with respect to a fixed Hermitian metric on \(X\)), then there exists a sequence \(\{r_n\}\subset\text{Aut}({\mathbb C})\) such that \(f_n\circ r_n\) admits a subsequence converging to a non-constant entire map \(f:{\mathbb C}\to X\). The original proof of Brody's lemma does not say anything on the localization of the entire curve \(f({\mathbb C})\) inside \(X\); the main result of this paper shows instead that we do have some control on it. The idea is the following: without loss of generality, one can assume that the length of \(\partial f_n(\Delta)\) is negligible with respect to the area of \(f_n(\Delta)\), and that the normalized integration currents \([f_n(\Delta)]/\text{Area}\bigl(f_n(\Delta)\bigr)\) converge toward a closed positive current \(T\). Then the author shows that if \(K\subset X\) is a compact set charged by \(T\), then we can choose the converging subsequence so that the entire curve \(f({\mathbb C})\) intersects \(K\) (and it is contained in the support of \(T\)). As a consequence, the author obtains a characterization of compact Kobayashi hyperbolic manifolds via an isoperimetric inequality: a compact complex manifold \(X\) is Kobayashi hyperbolic if and only if there is a constant \(C>0\) so that \(\text{Area}(D)\leq C\,\text{length}(\partial D)\) for all holomorphic disks \(D\subset X\).