We consider a $3$-dimensional complete Riemannian manifold $(M,g)$ with boundary and assuming that an optimal Sobolev or Hardy–Sobolev inequality holds with the same best constant as in the case of the solid torus $T$ we investigate the relationship between the geometry of such a manifold and that of the solid torus and we prove Ledoux-type rigidity results (see in Ledoux (1999)) giving a positive answer to a question posed to us by the referee of our recently published paper Cotsiolis and Labropoulos (2018). Some generalization results, for $n$-dimensional Riemannian manifolds $(M,g)$ with $n>3$, are also presented.

中文翻译：

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在存在对称性的情况下，带有边界的流形上出现Ledoux型刚度

我们考虑一个 $3$维完整黎曼流形 $（\mathrm{中号}，G）$ 有边界，并假设最优Sobolev或Hardy-Sobolev不等式与实心圆环的情况具有相同的最佳常数 $Ť$我们研究了这种歧管的几何形状与实体圆环的几何形状之间的关系，并证明了Ledoux型刚度结果（请参见Ledoux（1999））对我们最近发表的裁判提出的问题给出了肯定的答案论文Cotsiolis和Labropoulos（2018）。一些概括的结果，对于$ñ$维黎曼流形 $（\mathrm{中号}，G）$ 与 $ñ>3$，也介绍了。