Sharp Functional inequalities of hardy type and involving curvature on Riemannian manifolds

Abstract: In this talk, Hardy-type inequalities associated to the quadratic form of the shifted Laplacian $-\Delta _{\mathbb{H} N}-(N-1)^2 /4$ on the hyperbolic space $\mathbb{H}^N , (N-1)^2 /4$ being, as it is well-known, the bottom of the $L^2$-spectrum of $-\Delta _{\mathbb{H} N}$ will be presented. Sharpness of constants of the resulting Poincar\'e-Hardy inequality and the \textit{criticality} of the operator will also be discussed. Furthemore a related improved Hardy inequality on more general manifolds, under suitable \textit{curvature} assumption and allowing for the curvature to be possibly unbounded below, will be considered. It involves an explicit, curvature dependent and typically unbounded potential, and the resulting Schr\"odinger operator will be shown to be \textit{critical} in a suitable sense. If time permits I will also consider higher order analogue of improved Poincar\'e inequality and discuss related \textit{open questions}.