Some Dirichlet problems arising from conformal geometry (Q635673)

The main result of the paper states that every smooth compact manifold \(M\) (\(\dim \left( M\right) \geq 3\)) with boundary admits a complete smooth metric tensor \(g\) such that \(\sigma _{k}\left( g^{-1}\left( \text{Ric}_{g}- \text{R}_{g}g\right) \right) =\text{const}>0\) (\(k=1,2,\ldots,n\)), where \(\sigma _{k}\), \(k=1,2,\ldots,n\), denote the elementary symmetric functions and \(\text{Ric}_{g}\), \(\text{R}_{g}\) denote the Ricci and scalar curvature tensors, respectively; in particular, if \(k=n\), then \(\det \left( \text{Ric}_{g}- \text{R}_{g}g\right) =\text{const}>0.\) This result provides a generalization of results of \textit{P. Aviles} and \textit{R. C. McOwen} [Duke Math. J. 56, No.~2, 395--398 (1988; Zbl 0645.53023)]. The proof uses results of \textit{M. Gursky, J. Streets} and \textit{M. Warren} [Calc. Var. Partial Differ. Equ. 41, No.~1--2, 21--43 (2011; Zbl 1214.53035)].