2 problems

Kazdan–Warner's conjecture. When $Y(M)=0$ and $f\not\equiv0$, the equation possesses a positive solution if and only if

Let $\mathbb{H}^m_\kappa$ be hyperbolic space with curvature parameter $\kappa$, let $r(x)$ denote the distance from a fixed origin, and let $C>0$. A function $\widetilde s(x)$ is…