Nadirashvili surface

In differential geometry, a Nadirashvili surface is an immersed complete bounded minimal surface in R³ with negative curvature. The first example of such a surface was constructed by Nikolai Nadirashvili [de] in Nadirashvili (1996). This simultaneously answered a question of Hadamard about whether there was an immersed complete bounded surface in R³ with negative curvature, and a question of Eugenio Calabi and Shing-Tung Yau about whether there was an immersed complete bounded minimal surface in R³.

Hilbert (1901) showed that a complete immersed surface in R³ cannot have constant negative curvature, and Efimov (1963) show that the curvature cannot be bounded above by a negative constant. So Nadirashvili's surface necessarily has points where the curvature is arbitrarily close to 0.

References

Efimov, N. V. (1963), "The impossibility in Euclidean 3-space of a complete regular surface with a negative upper bound of the Gaussian curvature", Doklady Akademii Nauk SSSR, 150: 1206–1209, ISSN 0002-3264, MR 0150702
Nadirashvili, Nikolai (1996), "Hadamard's and Calabi–Yau's conjectures on negatively curved and minimal surfaces", Inventiones Mathematicae, 126 (3): 457–465, doi:10.1007/s002220050106, ISSN 0020-9910, MR 1419004