The geometry of Nadirashvili's complete, bounded, negative curvature surface

2011-01-22T18:20:14Z

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.

Answer by Tom Mrowka for The geometry of Nadirashvili's complete, bounded, negative curvature surface

2011-03-30T14:01:55Z

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.)

The geometry of Nadirashvili's complete, bounded, negative curvature surface - MathOverflow

The geometry of Nadirashvili's complete, bounded, negative curvature surface

Answer by Tom Mrowka for The geometry of Nadirashvili's complete, bounded, negative curvature surface