I would like to understand the geometric structure of
a surface that Nadirashvili constructed which resolved what
was known as Hadamard's Conjecture.
Perhaps in the 15 years since his construction, others have
redescribed the example, and perhaps even made a graphics image of it?
Background.
Hilbert's theorem that implies that the hyperbolic plane cannot
be realized as a surfaces in $\mathbb{R}^3$ is well known.
Perhaps less well known is Hadamard's Conjecture, which
asked if there is a complete negative curvature surface
in a bounded region of $\mathbb{R}^3$.
This is discussed at some length in Burago and Zallgaller's
book Geometry III: Theory of Surfaces. The problem was
solved after that 1989 book was written, as Berger explains
in A Panoramic View of Riemannian Geometry (p.135):
(Incidentally, the answer to this related MO question on Compact Surfaces of Negative Curvature
does not resolve my question, as it relies on Burago and Zallgaller.)
Here is the citation:
Nikolaj Nadirashvili,
"Hadamard's and Calabi-Yau's conjectures on
negatively curved and minimal surfaces."
Invent. Math. 126(3) (1996), 457–465.
The main theorem is this:
Theorem. There exists a complete surface of negative Gaussian curvature
minimally immersed in $\mathbb{R}^3$ which is a subset of the unit ball.
I have studied the paper, but my grasp of the underlying
mathematics is not strong enough to convert his description
into a geometric picture.
If anyone knows of later discussions that might help, I would
appreciate pointers or references. Thanks!
Edit. I was not able to access MathReviews until now. This is from the review by M. Cai (MR1419004 (98d:53014)):
For the proof, the author starts with a minimal immersion of the unit disk into a fixed ball in $\mathbb{R}^3$ with the Gaussian curvature of the immersed surface being negative, then he inductively defines a sequence of minimal immersions of negative curvature into the fixed ball in such a way that the sequence converges to a complete immersion.
This helps.
Best Answer
In fact the conjecture becomes true if you add embedded to the hypothesis according to a theorem of Colding and Minicozzi. (Colding, Tobias H.; Minicozzi, William P., II The Calabi-Yau conjectures for embedded surfaces. Ann. of Math. (2) 167 (2008), no. 1, 211–243.)
Who knows what Hadamard actually had in mind.
(Sorry this should have been a comment, its certainly not an answer to the question.)