论文标题
至少两端的流形上的非负标量曲率
Nonnegative scalar curvature on manifolds with at least two ends
论文作者
论文摘要
令$ m $为可定向连接的$ n $二维流形，$ n \ in \ {6,7 \} $，让$ y \ y \ subset m $是一个双面封闭的连接的不可压缩的超曲面，不承认一个积极的标量曲率（PSC缩写为PSC）。此外，假设$ m $和$ y $的通用封面既是旋转或两者旋转。使用Gromov的$μ$ bubbles，我们表明$ M $不承认PSC的完整指标。我们提供了一个示例，表明旋转/非自旋假设不能从此结果的陈述中删除。这是Gromov的大量案件的问题，最多达到$ 7 $。此外，我们证明了Condimension二的子手势的相关结果。 We deduce as special cases that, if $Y$ does not admit a metric of psc and $\dim(Y) \neq 4$, then $M := Y\times\mathbb{R}$ does not carry a complete metric of psc and $N := Y \times \mathbb{R}^2$ does not carry a complete metric of uniformly psc provided that $\dim(M) \leq 7$ and $ \ dim（n）\ leq 7 $。在可定位的歧管的情况下，这解决了尺寸$ 7 $的尺寸$ 7 $。
Let $M$ be an orientable connected $n$-dimensional manifold with $n\in\{6,7\}$ and let $Y\subset M$ be a two-sided closed connected incompressible hypersurface which does not admit a metric of positive scalar curvature (abbreviated by psc). Moreover, suppose that the universal covers of $M$ and $Y$ are either both spin or both non-spin. Using Gromov's $μ$-bubbles, we show that $M$ does not admit a complete metric of psc. We provide an example showing that the spin/non-spin hypothesis cannot be dropped from the statement of this result. This answers, up to dimension $7$, a question by Gromov for a large class of cases. Furthermore, we prove a related result for submanifolds of codimension two. We deduce as special cases that, if $Y$ does not admit a metric of psc and $\dim(Y) \neq 4$, then $M := Y\times\mathbb{R}$ does not carry a complete metric of psc and $N := Y \times \mathbb{R}^2$ does not carry a complete metric of uniformly psc provided that $\dim(M) \leq 7$ and $\dim(N) \leq 7$, respectively. This solves, up to dimension $7$, a conjecture due to Rosenberg and Stolz in the case of orientable manifolds.
