Classification of prime 3-manifolds with \(\sigma\) invariant greater than \(\mathbb RP^3\)
Let \(M^n\) be a smooth compact manifold of dimension \(n \geq 3\). Given a conformal class \([g]\) of Riemannian metrics, finding a metric in \([g]\) whose scalar curvature is constant is called the Yamabe problem. Many mathematicians contributed to solve this problem which finally is now completely solvable. It causes to study some conformal invariants and geometric invariants. Finding a constant scalar curvature metric is equivalent to compute the following conformal invariant \[ Y(M, [g]) = \inf_{\tilde g \in [g]} \frac{\int_M s_{\tilde g}\, dv_{\tilde g}} {\text{vol}(M, \tilde g)^{\frac{n-2}{n}}}, \] where \(s_{\tilde g}\) denotes the scalar curvature with respect to the metric \(\tilde g\). It is well-known that \(Y(M, [g]) \leq Y(S^n, [g_0])\), the conformal invariant of the standard \(n\)-sphere and so one can consider the mini-max procedure. The supremum of conformal invariants over all conformal classes is called the Yamabe constant which is a geometric invariant, and denoted by \(Y(M)\). The paper under review is related with these invariants for dimension \(n = 3\). In this case, one has \[ Y(M^3, [g]) = \inf_{\tilde g \in [g]} \frac{\int_M s_{\tilde g}\, dv_{\tilde g}}{ \text{vol}(M, \tilde g)^{\frac{1}{3}}} \] and \[ Y(M) \leq Y(S^3) = Y(S^3, [g_0]) = 6 \cdot (2\pi^2)^{\frac{2}{3}}. \] Recently there are some developments in computing the Yamabe constant and one of the main questions in this direction is that if \(\Gamma\) is a finite subgroup of \(O(4)\) acting freely on \(S^3\) with \(| \Gamma| = m\), then \[ Y(S^3/\Gamma) = \frac{Y(S^3)}{m^{\frac{2}{3}}}:=\sigma_m. \] In the paper the authors prove this conjecture to be true when \(m = 2\), that is, \(M = {\mathbb RP}^3\). More generally, they prove that if \(M\) is a closed \(3\)-manifold with \(Y(M) > \sigma_2\), then \(M\) is either \(S^3\), a connected sum with an \(S^2\) bundle over \(S^1\), or has more than one nonorientable prime component. As a corollary, they obtain \[ Y({\mathbb RP}^2 \times S^1) = Y({\mathbb RP}^3) = \sigma_2 \] and \(Y(M) = \sigma_2\) if \(M\) is a finite number of connect sums of \({\mathbb RP}^3\) and zero or one connect sums of \({\mathbb RP}^2 \times S^1\). The authors use the inverse mean curvature flow technique to prove their results. The corrigendum concerns the correction of the original title (\(\sigma\)-invariant replacing Yamabe invariant).
scientific article; zbMATH DE number 2123899