Prescribing almost constant scalar curvature and almost constant boundary mean curvature on B^n
DOI:
https://doi.org/10.12775/TMNA.2025.026Keywords
Conformal metrics, prescribed curvatures, Lapunov-Schmidt reductionAbstract
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 $\mathbf{K}$ and a boundary mean curvature which is equal to a given function $\mathbf{H}$ on the standard ball $(\mathbb B^n,g)$. Our analysis extends previous studies by considering the scenario where the curvatures $\mathbf{K}$ and $\mathbf{H}$ are close to constants $\mathbf{K}_0> 0$ and $\mathbf{H}_0> 0$. 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.References
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