Palais-Smale sequences for the prescribed Ricci curvature functional (Q6568728)

The prescribed Ricci curvature problem consists in finding, for a given \(2\)-tensor \(T\), a Riemannian metric \(g\) on a manifold \(M\) such that \N\[\N\mathrm{Ric}(g)=cT, \N\]\Nwhere \(c\) is a constant. Suppose that the metric \(g\) and the tensor \(T\) are invariant under a compact Lie group \(G\) acting on \(M\) and \(M\) is a homogeneous space \(G/H\). It is known (see [\textit{A. Pulemotov}, J. Geom. Phys. 106, 275--283 (2016; Zbl 1341.53080)]) that \(G\)-invariant metrics with Ricci curvature \(cT\) are, up to scaling, critical points of the scalar curvature functional \(S\) on the set \(\mathcal{M}_T = \mathcal{M}_T (G/H)\) of \(G\)-invariant metrics on \(G/H\) subject to the constraint \(tr_g T = 1\).\N\NMountain pass techniques require some form of compactness. The assumption imposed most commonly is the Palais-Smale condition. It postulates that every sequence of metrics \(g_i\) with \N\[\N\lim_ {i\to\infty} S(g_i) = \lambda \in \mathbb{R}\text{ and }\lim_ {i\to\infty} | \mathrm{grad}\, S_{|\mathcal{M}_T}(g_i)|_{g_i} = 0\N\]\Nhas a convergent subsequence.\N\NThe main result of the paper classifies divergent Palais-Smale sequences for the functional \(S|_{\mathcal{M}_T}\). As an application, existence of saddle points is proved on generalized Wallach spaces and several types of generalised flag manifolds. Also the image of the Ricci map is described in some examples.