Prescribing almost constant scalar curvature and almost constant boundary mean curvature on B^n - Pu, Yixing (Topological Methods in Nonlinear Analysis, 2025)
In this paper, we investigate a boundary case of the classical prescribed curvature problem. We focus on a prescribing scalar curvature which is equal to a given function $K$ and a boundary mean curvature which is equal to a given function $H$ on the standard ball $(\mathbb B^n,g)$. Our analysis extends previous studies by considering the scenario where the curvatures $K$ and $H$ are close to constants $K_0> 0$ and $H_0> 0$. Using a perturbative approach and leveraging the ansatz introduced by Han-Li \cite{HanLi_2000}, we establish new existence results for the conformal metric when the prescribed curvatures are near constants.