Conformal geometry, contact geometry, and the calculus of variations
Let \((N,g_0)\) be a compact Riemannian manifold of dimension \(n\geq 3\). In this paper, the author considers the curvature equations \[ \sigma_k\left( \text{Ric}_g-{R_g\over 2(n-1)} \cdot g\right)= \text{constant} \tag{1} \] for metrics \(g\) in the conformal class of \(g_0\), where \(\text{Ric}_g\) and \(R_g\) are the Ricci tensor and the scalar curvature of the metric \(g\), respectively, and where \(\sigma_k\) is the \(k\)-th elementary symmetric function of the eigenvalues \((g\) is used to view \((2,0)\)-tensors as endomorphisms of the tangent bundle). The case \(k=1\), \(R_g=\) constant is known as the Yamabe problem. The first theorem shows that unit volume solutions of (1) are critical points of the functional \[ {\mathcal F}_k:g \mapsto\int_N \sigma_k\left( \text{Ric}_g-{R_g\over 2(n-1)} \cdot g\right)d V_g\tag{2} \] when \((N,[g])\) is locally conformally flat and \(k\neq n/2\). If \(N\) is not locally conformally flat, the statement is true for \(k=1,2\). From this result, it follows that, for all \(k\), a positive scalar curvature Einstein metric is a strict local minimum unless \((N,[g_0])\) is conformally equivalent to the standard unit sphere \(S^n\), in which case there is an \((n+1)\)-parameter family of local minima. If \(N\) is not locally conformally flat, again the statement holds for \(k=1,2\). Further, if \((N,g_0)\) is of unit volume and constant sectional curvature \(K\neq 0\), then it is proven that, for \(1\leq k<n-1\), \(g_0\) is the unique unit volume solution of (1) in its conformal class unless \((N,g_0)\) is isometric to the \(S^n\) with the standard metric. In this case there is an \((n+1)\)-parameter family of solutions obtained as images of the standard metric under the conformal transformations of \(S^n\). The methods used are geometric and employ relations between conformal geometry and contact geometry to show the correspondence between the solutions of (1) and the critical points of (2). The characterization of the solutions is obtained by examining the behavior of the functional \({\mathcal F}_k\) near a critical point using the second variation and again employing the relations derived from conformal geometry and contact geometry. As for the uniqueness of unit volume solutions, these methods do not work for the case \(k=n\). Nevertheless, the result for the case where \(K<0\) is established for all \(k\) using a maximum principle argument. The author provided a partial differential equation proof for all \(k\) also when \(K> 0\) in [Trans. Am. Math. Soc. 352, 4371-4379 (2000; Zbl 0951.35044)]; for an outline of the proof see also [Some fully nonlinear equations in conformal geometry. Differential equations and mathematical physics (Birmingham, AL, 1999), AMS/IP Stud. Adv. Math. 16, 425-433 (2000; Zbl 1161.53346)].
scientific article; zbMATH DE number 1442144
scientific article