Prescribing curvature problem of Bakry-Émery Ricci tensor (Q382767)

Let \( (M,g)\) be a connected Riemannian manifold of dimension \(n\geq3\). Let \(f\in C^\infty(M)\). The Bakry-Émery Ricci tensor is given by \(\mathrm{Ric}_f:=\mathrm{Ric}+\mathrm{Hess}(f)\). Let \(\sigma_k\) be the \(k\)th elementary symmetric function of the eigenvalues. Then \(\sigma_k(\mathrm{Ric}_f)\) is a smooth function on the manifold. It is natural to normalize this by considering \(\sigma_k(\mathrm{Ric}_f)^{1/k}\); this gives rise to the \(k\)-curvature. The author considers the problem of deforming the given metric \(g\) within its conformal class so that the \(k\)-curvature is constant; this is a generalization of the Yamabe problem which corresponds to \(k=1\) and \(f=0\). The author shows that this problem can be solved provided that the initial curvature \(\mathrm{Ric}_f\) belongs to a suitable negative cone. He also considers the associated Monge-Ampere equation. The first section contains a statement of the problem and a historical outline, the second section deals with ellipticity and maximum principle, the third, fourth, and fifth sections deal with \(C^0\), \(C^1\), and \(C^2\) estimates, respectively, and the final section establishes the desired result.