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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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References:
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