On the problem of realization of a given Gaussian curvature function
The author investigates the local realization problem for the Gaussian curvature of a surface: Given a germ of a smooth function of two variables, find a surface whose Gaussian curvature is the given function. It is well known that any function germ \(g(x,y)\) is realizable as the Gaussian curvature of a surface \(z=f(x,y)\) if the curvature value at the central point is not zero, or if the curvature is vanishing at the central point but its differential does not vanish. The main result is, that the problem can be solved whenever \(g\) has a critical point of finite multiplicity at the origin. Similar results are proved for the local realization problem for the Hessian. The proofs proceed by solving the problem in the setting of formal power series.
scientific article; zbMATH DE number 1270098