Spectral theory of a conformal class (Q1908861)

On a compact manifold the study of the conformal class \(C(g)\) of a Riemannian metric \(g\) consists of the determination of all metrics which possess a given property and of the determination of conformal invariants. In this paper the author studies the case of metrics with a given constant scalar curvature for a compact manifold \((V, g)\) of dimension \(n\geq 3\). Following the approaches of Yamabe, Aubin, Schoen, if \(C(g)\) is the set of the metrics \(g'\) with \(g'= \varphi^{4/(n- 2)} g\), the value of the scalar curvature \(R'\) of \(g'\) is a given constant \(\nu\) iff \(\varphi\) verifies the Yamabe equation \[ {4(n-1)\over n-2}\Delta\varphi+ R\varphi= \nu\varphi^{(n+ 2)/(n- 2)} \tag{1} \] where the solution can be normalized in such a way that \[ \int \varphi^p dV= 1,\tag{2} \] with \(p= 2n/(n-2)\). The spectrum \(\text{sp}(C)\) of the conformal class \(C\) is then defined as the set of all values \(\nu\) such that there exists a solution of (1) satisfying (2), and \(S_\nu\) indicates the set of all conformal metrics with scalar curvature \(\nu\) and volume 1. Moreover, if \(\mu(C)= \inf_{g'\in C} E(g')\) where \(E(g')\) is the functional energy of the conformal metric \(g'\), from results of Aubin, Yamabe, Schoen follows \(\mu(C)\in \text{sp}(C)\), and all the metrics of \(S_{\mu(C)}\) are called minimal Yamabe solutions. From the study of the Yamabe equation, the author proves that \(S(C)\) has the structure of an analytic set. He gives the cases in which the Yamabe problem has a unique solution and when a family of solutions corresponding to a family of metrics is relatively compact.