Conformal complete metrics with prescribed non-negative Gaussian curvature in \({\mathbb{R}}^ 2\)
The paper deals with the existence of (complete) Riemannian metrics g on \({\mathbb{R}}^ 2\), conformal to the Euclidean metric \(g_ 0\) (by \(g=e^{2u}g_ 0)\) and possessing a prescribed Gaussian curvature \(k: {\mathbb{R}}^ 2\to {\mathbb{R}}\). This problem is intimately related to the nonlinear elliptic PDE \(\Delta u+k(x)e^{2u}=0\). It is always assumed that k is Hölder-continuous and that k(x) for large \(| x|\) is asymptotically comparable to some negative power of \(| x|\), in one sense or the other. For example, if k is positive somewhere and, asymptotically, \(| k(x)| \leq M/| x|^ b\) for some constants \(M>0\), \(b\geq 2\) then there is a complete metric g, for which an asymptotic rule like \[ (b-c)\cdot \ln | x| -2\tilde C\leq 2u(x)\leq (b-c)\cdot \ln | x| +2\tilde C \] holds. In case \(k\geq 0\) the condition \(b\geq 2\) may be relaxed to \(b>0\), provided some other assumptions are correspondingly sharpened. This is the most complicated case considered here. Several situations are exhibited where every solution of the elliptic PDE has an asymptotic behaviour like this. Finally, it is shown in the present frame that Cohn-Vossen's inequality is necessary and sufficient for the metric g to become complete. The methods are mainly analytical in nature, including e.g. the Leray- Schauder fixed point theory and the capacity of planar sets.
scientific article; zbMATH DE number 3951482
scientific article