Hardy and Hardy-Sobolev inequalities on Riemannian manifolds

Let $(M,g)$ be a smooth compact Riemannian manifold of dimension $N\ge 3$. Given $p_0\in M$, $\lambda \in \mathcal{R}$ and $\sigma \in (0,2]$, we study existence and non existence of minimizers of the following quotient: \begin{equation}\label{Paper Equation} \mu_{\lambda,\sigma}=\inf_{u \in H^1(M)\setminus \lbrace0\rbrace} \frac{\displaystyle\int_M |\nabla u|^2 dv_g -\lambda \int_M u^2 dv_g }{\biggl(\displaystyle\int_M \rho^{-\sigma} |u|^{2^(\sigma)} dv_g\biggl)^{2/2^(\sigma)}}, \end{equation} where $\rho (\mathrm{.}):=dist(p_0,\mathrm{.})$ denoted the geodesic distance from $p\in M$ to $p_0$. In particular for $\sigma =2$, we provide sufficient and necessary conditions of existence of minimizers in terms of $\lambda$. For $\sigma \in (0,2)$ we prove existence of minimizers under scalar curvature pinching.