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On Nirenberg’s problem. (English) Zbl 0786.58010
The paper focusses on the problem of characterizing the functions \(K\) which can be the Gaussian curvature of a metric \(g\) on \(S^ 2\) which is pointwise conformal to the standard metric \(g_ 0\) (Nirenberg’s Problem).
The authors face the problem through its reduction to the solvability of a partial differential equation which is the Euler-Lagrange equation of a functional and to the study of its critical points using their generalization of Morse theory to general boundary conditions. By avoiding complicated techniques and relying on more conceptual arguments, they claim a simplification of the usual treatment of the problem. Besides obtaining further results, the authors claim to have all known results in the literature unified under their proof.

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
58E11 Critical metrics
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