Let $B(\delta)\subset\mathbb S^n$ be a geodesic ball with the round metric $\bar{g}$ of sectional curvature one, where $n\geq3$ and $0<\delta<\pi/2$. We prove local scalar-curvature rigidity under preservation of the induced boundary metric and a one-sided lower bound for the boundary mean curvature. More precisely, ev...
We prove sharp residual stability estimates for the critical Poincar\'e-Sobolev equation near sums of one positive hyperbolic solution and one projected Euclidean bubble in dimensions at least four. The estimates are uniform in the relative positions of the centers. The sharp rate is linear except in dimension six, whe...
We determine the sharp compactness ranges for the scalar-flat and minimal-boundary Yamabe problems on smooth compact manifolds of positive conformal type, excluding the conformal round hemisphere. For zero scalar curvature and positive constant boundary mean curvature, compactness holds through dimension $14$ for gener...
Liu-Wei Gong, Seunghyeok Kim, M. Musso et al.· 0 citations
We construct smooth, non-locally-conformally-flat metrics on the closed ball for which the boundary Yamabe equation admits $L^\infty$-unbounded sequences of positive solutions. The background metric and the prescribed scalar and boundary mean curvatures are fixed along each sequence. For zero scalar curvature and posit...
Liu-Wei Gong, Seunghyeok Kim, M. Musso et al.· 0 citations
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