  We prove a boundedness-theorem for families of abelian varieties with real
multiplication. More generally, we study curves in Hilbert modular varieties
from the point of view of the Green Griffiths-Lang conjecture claiming that
entire curves in complex projective varieties of general type should be
contained in a proper subvariety. Using holomorphic foliations theory, we
establish a Second Main Theorem following Nevanlinna theory. Finally, with a
metric approach, we establish the strong Green-Griffiths-Lang conjecture for
Hilbert modular varieties up to finitely many possible exceptions.
