The $\sigma_k$-Yamabe problem revisited

In this paper we revisit the $\sigma_k$-Yamabe problem on $M^n$, namely, finding a conformal metric with constant $\sigma_k$-scalar curvature. We prove that on a closed manifold $\left(M,\left[g_0\right]\right)$ with positive Yamabe constant $Y_1\left(M,\left[g_0\right]\right)>0$, the $\sigma_2$-Yamabe constant $$Y_2\left(M,\left[g_0\right]\right):=\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}$$ is achieved by a conformal metric $g \in\left[g_0\right]$, which in particular solves the $\sigma_2$-Yamabe problem, assuming $Y_2\left(M,\left[g_0\right]\right)>0$. As a consequence, for any $\left(M, g_0\right)$ with $Y_1\left(M,\left[g_0\right]\right)>$ 0 and $Y_2\left(M,\left[g_0\right]\right)>0$ one has $$\inf _{g \in\left[g_0\right], R_g>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}}=\inf _{g \in\left[g_0\right], R_g>0, \sigma_2(g)>0} \frac{\int_M \sigma_2(g) d \operatorname{vol}(g)}{\operatorname{vol}(g)^{\frac{n-4}{n}}} .$$ We also show that these conclusions can fail if the condition $R_g>0$ is removed.