Nirenberg's problem in 90's (Q2735670)

Part of the research of 2-dimensional Riemannian geometry is the study of Nirenberg's problem. The author considers a standard unit 2-sphere in Euclidean space \(R^3\). Then Nirenberg's problem is to find functions \(K(x)\) on \(S^2\) such that they are the Gaussian curvature functions of metrics \(g\) which are pointwise conformally equivalent to \(c\) on \(S^2\). The problem is to find a smooth function \(u:S^2\to\mathbb{R}^2\) satisfying the equation \(\Delta u=i-Ke^{2u}\). In this case the metric \(g=e^{2u}c\) is conformal to \(c\) and \(K\) is the Gaussian curvature. This note contains two parts: the existence and compactness.NEWLINENEWLINENEWLINEFor other details, see the author's references and \textit{J. L. Kazdan}, Prescribing the curvature of a Riemannian manifold, Reg. Conf. Ser. Math. 57 (1985; Zbl 0561.53048).NEWLINENEWLINEFor the entire collection see [Zbl 0954.00038].