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Prescribing Gaussian curvature on \(S^ 2\). (English) Zbl 0715.53034
What functions K can be the Gaussian curvature of a metric on \(S^ 2\) which is pointwise conformal to the standard metric? It is well known that this problem reduces to solving the nonlinear PDE \(\Delta u=1- Ke^{2u}\) where \(\Delta\) is the Laplacian on \(S^ 2\) with its standard metric.
The principal results of this paper concern the case when all the critical points P of K with \(K(P)>0\) are nondegenerate and satisfy \(\Delta\) \(K\neq 0\). If p is the number of positive local maxima and q is the number of positive saddle points with \(\Delta K<0\) then there exists a solution if p-q\(\neq 1\). If furthermore \(K>0\), then the number of solutions can be estimated and a priori estimates of the solution of the PDE can be given.
These results depend on a generalized Morse theory for a corresponding variational problem.
Reviewer: J.J.Hebda

MSC:
53C20 Global Riemannian geometry, including pinching
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
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