I know that the pseudosphere is a regular surface with Gaussian curvature $-1$ that is not complete, also this surface is not complete. Hilbert's theorem ensures that there is no isometric immersion of the hyperbolic plane $\mathbb{H}^2$ into $\mathbb{R}^3$, but if we remove a point from the hyperbolic plane, can it be immersed isometrically in $\mathbb{R}^3$? What could it be?

I read this: Dini's surface provides a analytic isometric embedding of the hyperbolic plane. That region includes hyperbolic disks of any finite radius, though big disks will be wrapped many times, very tightly around the screw axis of Dini's surface. But how could I prove it? I have already tried a few days without success.

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