Nadirashvili surface

In differential geometry, a Nadirashvili surface is an immersed complete bounded minimal surface in ${\displaystyle \mathbb {R} ^{3}}$ with negative Gaussian curvature. The first example of such a surface was constructed by Nikolai Nadirashvili [de] in 1996. This simultaneously answered a question of Hadamard about whether there was an immersed complete bounded surface in ${\displaystyle \mathbb {R} ^{3}}$ with negative Gaussian curvature, and a question of Eugenio Calabi and Shing-Tung Yau about whether there was an immersed complete bounded minimal surface in ${\displaystyle \mathbb {R} ^{3}}$.^[1]

Hilbert (1901) showed that a complete immersed surface in ${\displaystyle \mathbb {R} ^{3}}$ cannot have constant negative Gaussian curvature,^[2] and Efimov (1963) show that the curvature cannot be bounded above by a negative constant.^[3] Therefore, Nadirashvili's surface necessarily has points where the Gaussian curvature is arbitrarily close to 0. As a minimal surface, its mean curvature is 0 everywhere. Topologically, it is a disk. As an immersed surface, it intersects itself; it is not embedded. These self-intersections are necessary, as Colding and Minicozzi proved in 2008 that embedded complete bounded minimal disks do not exist.^[4]

References

1. 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
2. Hilbert, David (1901), "Ueber Flächen von constanter Gaussscher Krümmung" [On surfaces of constant Gaussian curvature], Transactions of the American Mathematical Society (in German), 2 (1), American Mathematical Society (AMS): 87–99, doi:10.1090/s0002-9947-1901-1500557-5, JSTOR 1986308, MR 1500557
3. 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
4. Colding, Tobias H.; Minicozzi, William P., II (2008), "The Calabi-Yau conjectures for embedded surfaces", Annals of Mathematics, Second Series, 167 (1): 211–243, doi:10.4007/annals.2008.167.211, MR 2373154