One way to generalize the boundary Yamabe problem posed by Escobar is to ask if a given metric on a compact manifold with boundary can be conformally deformed to have vanishing ${\sigma }_k$-curvature in the interior and constant $H_k$-curvature on the boundary. When restricting to the closure of the positive k-cone, this is a fully nonlinear degenerate elliptic boundary value problem with fully nonlinear Robin-type boundary condition. We prove nonuniqueness for the boundary-value problem ${\sigma }_4$-curvature equals zero and constant $H_4$-curvature by using bifurcation results proven by Case, Moreira and Wang. Surprisingly, our construction via products of sphere and hyperbolic space only works for a finite set of dimensions.

概括由Escobar提出的边界Yamabe问题的一种方法是，询问具有边界的紧凑流形上的给定度量是否可以共形变形以消失 ${\sigma }_{ķ}$内部曲率恒定 $H_{ķ}$边界上的曲率。当限制正k-锥的闭合时，这是具有完全非线性Robin型边界条件的完全非线性简并椭圆边界值问题。我们证明了边值问题的非唯一性${\sigma }_4$-曲率等于零且恒定 $H_4$通过使用Case，Moreira和Wang证明的分叉结果获得曲率。出人意料的是，我们通过球面和双曲空间乘积进行的构造仅适用于有限的一组尺寸。