an:00279820
Zbl 0786.58010
Chang, Kung Ching; Liu, Jia Quan
On Nirenberg's problem
EN
Int. J. Math. 4, No. 1, 35-58 (1993).
00014506
1993
j
58E05 58E11
Nirenberg's Problem; Gaussian curvature; Euler-Lagrange equation; critical points; Morse theory
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.
U.D'Ambrosio (Campinas)