Some local and nonlocal variational problems in Riemannian geometry (Q2765848)

This paper represents a brief summary of some recent work of the author with \textit{J. Viaclovsky} on two variational problems in Riemannian geometry [Invent. Math. 145, No. 2, 251-278 (2001; Zbl 1006.58008)]. Let \(M\) be a smooth manifold, \(\mathcal M\) be the space of smooth Riemannian metrics on \(M\), \(\mathcal M_1 = \{g\in\mathcal M: \text{vol}(g) = 1\}\). In the case of dimension3, \(R_g\) denotes the scalar curvature of \(g\), \(C_g=\text{Ric}-Rg/4\), \(\mathcal F_2[g]=\int_M\sigma_2(C_g) d \text{vol}_g\) the Riemannian functional, where \(\sigma_k:\mathbb R^3\to \mathbb R\) denotes the elementary symmetric function. NEWLINENEWLINENEWLINEThe following new characterization of a compact Einstein three-manifold is given: Let \(M\) be compact and three-dimensional. Then a metric \(g\) with \(\mathcal F_2[g]\geq 0\) is critical for \(\mathcal F_2|_{\mathcal M_1}\) if and only if \(g\) has constant sectional curvature. A metric \(g\) with \(\mathcal F_2[g]=0\) is critical for \(\mathcal F_2|_{\mathcal M_1}\) iff \(R\leq 0,\) \(g\) is locally conformally flat, and the eigenvalues of the tensor \(C_g\) are \(\{0,0,R/4\}\). A constrained version of the problem is also considered. Let \(\Xi=\{g\in\mathcal M_1: \int_M\sigma_2(C_g) d \text{vol}_g>0\), and \(R_g<0\}\). If \(g\) is a critical point of \(\mathcal F_2|_{\Xi}\), then \(g\) is hyperbolic.NEWLINENEWLINEFor the entire collection see [Zbl 0973.00041].