Surfaces in \(\mathbb{R}^ 3\) with prescribed curvature
The author studies surfaces in 3-dimensional Euclidean and 3-dimensional Minkowski space. His main result is a relation of the derivatives of the surface immersion, the Gauß curvature, and the derivative of the Gauß map. To me, this relation seems to be a re-expression of the classical fact that the Gauß curvature is the determinant of the Weingarten map. He then writes down integrability conditions for the surface in terms of the Gauß curvature and the Gauß map. He notes that under certain specializations these integrability conditions are equivalent to the Ernst equation in general relativity which governs stationary, axisymmetric spacetimes whose Killing vector fields commute.
scientific article; zbMATH DE number 839167
scientific article