论文标题
广义的肥皂泡和具有正标曲率的歧管的拓扑结构
Generalized soap bubbles and the topology of manifolds with positive scalar curvature
论文作者
论文摘要
我们证明，对于$ n \ in \ {4,5 \} $，一个封闭的非球体$ n $ -Manifold不承认具有正标曲率的Riemannian度量。 此外，我们表明，对于$ n \ leq 7 $，带有任意歧管的$ n $ torus的连接总和不承认完全的正标度曲率指标。当结合Lesourd-unger-Yau的贡献时，这证明了Schoen-Yayau Liouville定理为所有本地固定的平坦流形带有非负标态曲率。 这些结果中的一个关键工具是通用的肥皂气泡 - 对于规定的均曲面功能（也称为$μ$ bubbles）的固定的表面。
We prove that for $n\in \{4,5\}$, a closed aspherical $n$-manifold does not admit a Riemannian metric with positive scalar curvature. Additionally, we show that for $n\leq 7$, the connected sum of a $n$-torus with an arbitrary manifold does not admit a complete metric of positive scalar curvature. When combined with forthcoming contributions by Lesourd--Unger--Yau, this proves that the Schoen--Yau Liouville theorem holds for all locally conformally flat manifolds with non-negative scalar curvature. A key tool in these results are generalized soap bubbles -- surfaces that are stationary for prescribed-mean-curvature functionals (also called $μ$-bubbles).
