On the \(C^0\) compactness of the set of the solutions of the Yamabe equation (Q1004530)

The author investigates the compactness of the set of solutions of the following equation (Yamabe type equation): \[ \triangle\varphi +{{n-2}\over{4(n-1)}}R\varphi=n(n-2)\varphi^{(n+2)(n-2)}\tag{\(\clubsuit\)} \] where \(\varphi\) is a positive numerical function on a smooth compact Riemannian manifold of dimension \(n\), and \(R\) is its scalar curvature. (The note directly refers to some previous papers on the same subject, some of them are by the same author.) The Yamabe equation was introduced in relation with an attempt to prove the Poincaré conjecture. This equation encodes the problem to find on a compact smooth Riemannian manifold \((M,g)\) a metric \(g'=\varphi g\), i.e., conformal to \(g\), such that the corresponding scalar curvature \(R'\) is constant. Improvements of the solution proposed by the same Yamabe in 1960, (after N. Trubinger in 1968 showed a gap in Yamabe's proof), was given just by Thierry Aubin in 1976, by using functional methods in variational calculus. (The Yamabe equation is of variational type with respect to a suitable functional \(J(\varphi)\).) This method is related to the socalled Yamabe number \(\mu(M,g)\) of \((M,g)\), that is the infimum of the functional \(J(\varphi)\) evalued on a suitable subset of the Sobolev space \(H_1(M)\). More precisely, it is related to the inequality \(\mu(M,g)\leq \mu(S^n,g_0)\equiv n(n-1)\text{vol}(S^n,g_0)\), where \((S^n,g_0)\) is the \(n\)-dimensional sphere of radius \(1\). The strict inequality for manifolds not conformally equivalent to the sphere was conjectured also by Aubin and next proved by Schoen and Yau in 1988. The main result in this note is the following theorem. Theorem. When the conformal Laplacian is invertible on a smooth compact Riemannian manifold, not conformal [not conformally equivalent] to the canonical sphere. The set of solutions of the equation \((\clubsuit)\) is compact in \(C^k\), for any \(k\in {\mathbb N}\). Reviewer's remark. Nowadays, there is a lot of mathematical literature devoted to study different aspects of the Yamabe type equations. In general, such studies utilize functional analysis methods. However, more recently new geometric techniques are also considered, in the framework of the geometric theory of PDEs introduced by the reviewer of this note. In particular, in the paper [\textit{R. P. Agarwal} and \textit{A. Prástaro}, Adv. Math. Sci. Appl. 17, No.~1, 267--285 (2007; Zbl 1140.53005)] some interesting problems in Riemannian geometry are considered. It appears clear there that the Yamabe equation is not a suitable equation to use in order to prove the Poincaré conjecture. To this purpose the Ricci flow equation is instead the right one.