Conformal metrics with prescribed curvature on \(S^ n\)
If \(ds^ 2_ 0\) is a standard metric on \(S^ n\) considering a conformal change of it of the type \(ds^ 2=e^{2u}ds^ 2_ 0\) for \(n=2\) and \(ds^ 2=u^{4\over n-2}\) for \(n\geq 3\), the Gaussian curvature \((n=2)\) and scalar curvature \((n\geq 3)K\) of the new metrics verifies: \((1)\quad \triangle u+Ke^{2u}=1\text{ on } S^ 2 \)\qquad\( (2)\quad {4(n-1)\over n-2} \triangle u+Ku^{{n+2\over n-2}}=n(n-1)u\), \(n\geq 3\) where \(\triangle\) denote the Laplacian relative to the standard metric \(ds_ 0^ 2\). Several authors beginning with L. Nirenberg, J. Moser a.s.o. raised the inverse problem: which functions \(K\) can be the Gaussian curvature or scalar curvature of a metric \(ds^ 2\) which is a conformal change of \(ds^ 2_ 0\)? The authors study the equation (2) where \(K\) has a rotational symmetry. There are briefly presented the results of the article: \textit{K.-S. Cheng} and \textit{J. A. Smoller} [Conformal metric with prescribed Gaussian curvature on \(S^ 2\), Trans. Am. Math. Soc. (in press)] elaborating a comparative study for those of \(S^ n\) \(n\geq 3\). They demonstrate a necessary and sufficient condition under which the equation (2) have a regular solution and four types of ODE approaches in which we have a positive solution, which constitutes a new result and extends the obtained results by \textit{C.-S. Lin} and \textit{S.-S. Lin} [Appl. Anal. 38, No. 3, 121-159 (1990; Zbl 0708.35035)].
scientific article; zbMATH DE number 492696