Hilbert’s Theorem states that no complete surface of constant negative curvature can be isometrically immersed in $\mathbb R^3$. For appropriate choice of parameters, Dini’s surface has curvature $-1$, infinite area, and contains an arbitrarily large embedded hyperbolic disk.
While these two facts are not at odds, I’m having trouble understanding why Hilbert's proof of his eponymous Theorem cannot be extended to prohibit the existence of Dini's surface. In particular, Hilbert's proof shows that the coordinate curves of a particular parametrization of a curvature $-1$ surface (the one formed by its asymptotic curves) has the property that any coordinate rectangle has area less than $2\pi$. My understanding is that this would be true at least locally on Dini's surface. But if parameters are chosen for Dini's surface so that it contains a hyperbolic disk of radius, say, $100$, then surely that disk contains a coordinate rectangle of this parametrization of area $2\pi$.
What am I missing here?
I now understand what I was missing. The parametrization of Dini's surface whose coordinate curves are asymptotic curves has the property that these coordinate curves are nearly parallel near the axis of the surface. Therefore coordinate rectangles may have a very large diameters but very small areas. In particular, if a copy of Dini's surface has parameters such that it has curvature $-1$ and contains a hyperbolic disk $\mathcal D$ of radius $100$, then all of the coordinate rectangles are so long and skinny that any one of them that is a subset of $\mathcal D$ has area less than $2\pi$.