Nadirashvili surface
# Nadirashvili surface

In differential geometry, a Nadirashvili surface is an immersed complete bounded minimal surface in $R^3$ 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 $R^3$ with negative curvature, and a question of Eugenio Calabi and Shing-Tung Yau about whether there was an immersed complete bounded minimal surface in $R^3.$

Hilbert (1901) harvtxt error: no target: CITEREFHilbert1901 (help) showed that a complete immersed surface in $R^3$ 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.

