论文标题
带正标曲率的Riemannian自旋歧管的长颈原理
A long neck principle for Riemannian spin manifolds with positive scalar curvature
论文作者
论文摘要
在拓扑信息是由远离边界支撑的束束编码的情况下，我们开发了与边界紧凑型黎曼自旋歧管有关的索引理论。 As a first application, we establish a "long neck principle" for a compact Riemannian spin $n$-manifold with boundary $X$, stating that if $\textrm{scal}(X)\geq n(n-1)$ and there is a nonzero degree map into the sphere $f\colon X\to S^n$ which is strictly area decreasing, then the distance between the support of $ \ textrm {d} f $和$ x $的边界最多是$π/n $。这是Gromov最近提出的一个问题，在旋转设置和严格区域减少地图中的答案。作为第二个应用程序，我们考虑通过删除$ k $ k $成对的脱节嵌入$ n $ n $ ball的Riemannian歧管$ x $，从封闭的自旋$ n $ n $ -manifold $ y $中获得。我们表明，如果$ \ textrm {scale}（x）>σ> 0 $和$ y $满足以较高索引理论表示的一定条件，那么$ \ partial x $的地理领邻域的半径最多是$π\ sqrt {（n-1）/（nσ）} $。最后，我们考虑了Riemannian $ n $ -Manifold $ v $ diffemorphic至$ n \ times [-1,1] $的情况，$ n $ a $ n $ a封闭的旋转歧管带有不变的Rosenberg Index。在这种情况下，我们表明，如果$ \ textrm {scal}（v）\geqσ> 0 $，则$ v $的边界组件之间的距离最多为$2π\ sqrt {（n-1）/（nσ）} $。由于格罗莫夫（Gromov）引起的争论，这个最后一个常数是尖锐的。
We develop index theory on compact Riemannian spin manifolds with boundary in the case when the topological information is encoded by bundles which are supported away from the boundary. As a first application, we establish a "long neck principle" for a compact Riemannian spin $n$-manifold with boundary $X$, stating that if $\textrm{scal}(X)\geq n(n-1)$ and there is a nonzero degree map into the sphere $f\colon X\to S^n$ which is strictly area decreasing, then the distance between the support of $\textrm{d} f$ and the boundary of $X$ is at most $π/n$. This answers, in the spin setting and for strictly area decreasing maps, a question recently asked by Gromov. As a second application, we consider a Riemannian manifold $X$ obtained by removing $k$ pairwise disjoint embedded $n$-balls from a closed spin $n$-manifold $Y$. We show that if $\textrm{scal}(X)>σ>0$ and $Y$ satisfies a certain condition expressed in terms of higher index theory, then the radius of a geodesic collar neighborhood of $\partial X$ is at most $π\sqrt{(n-1)/(nσ)}$. Finally, we consider the case of a Riemannian $n$-manifold $V$ diffeomorphic to $N\times [-1,1]$, with $N$ a closed spin manifold with nonvanishing Rosenberg index. In this case, we show that if $\textrm{scal}(V)\geqσ>0$, then the distance between the boundary components of $V$ is at most $2π\sqrt{(n-1)/(nσ)}$. This last constant is sharp by an argument due to Gromov.
