On the curvature of nonregular saddle surfaces in the hyperbolic and spherical three-space
It is possible to characterise a saddle surface in Euclidean space by how it intersects hyperplanes: it is impossible to cut off a `crust' with a hyperplane. In the case of regular surfaces (say of class \(C^2\)), the Gaussian curvature of a saddle surface is necessarily nonpositive. However, it is possible to consider nonregular surfaces defined as the graph of a continuous function, for example. The notion of saddle surface still makes sense. It is also possible to define an intrinsic curvature known as curvature in the sense of Aleksandrov. This is a metric space notion defined by comparing small geodesic triangles with triangles having the same edge lengths in a space of constant curvature. With these notions, it is known for nonregular surfaces in Euclidean 3-space that, once again, the intrinsic curvature is nonpositive. In higher dimensions, however, the nonregular result is not known. In this article, the author extends the 3-dimensional Euclidean result to nonregular surfaces in hyperbolic 3-space or in the 3-sphere. In this case, the Aleksandrov curvature of a saddle surface is bounded above by the curvature of the ambient space.
scientific article; zbMATH DE number 1780526