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arXiv 2002.07131math.DGmath.KT
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具有正数量曲率的黎曼自旋流形的长颈原理

A long neck principle for Riemannian spin manifolds with positive scalar curvature

Simone Cecchini

备注：
Revised version to appear in GAFA. In Theorem A, the map f is strictly area decreasing, but see Remark 1.7. Technical issues fixed in the proofs of Lemma 5.5 and Theorem 5.2. The inequality in Proposition 3.8 is slightly strengthened
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摘要：

在拓扑信息由远离边界支撑的丛编码的情况下，我们发展了带边界的紧致黎曼自旋流形上的指数理论。作为第一个应用，我们建立了具有边界$X$的紧致黎曼自旋$n$ -流形的“长颈原理”，指出如果$\textrm{scal}(X)\geq n(n-1)$且存在一个到球面$f\colon X\to S^n$的非零度映射且该映射严格面积递减，则$\textrm{d} f$的支撑到$X$的边界之间的距离至多为$π/n$ .这回答了，在自旋设置和严格的面积减少的地图，一个问题，最近提出的格罗莫夫。作为第二个应用，我们考虑一个黎曼流形$X$通过删除$k$成对不相交嵌入$n$ -球从一个封闭的自旋$n$ -流形$Y$。证明了如果$\textrm{scal}(X)>σ>0$和$Y$满足一定的高指标条件，则$\partial X$的测地领邻域的半径至多为$π\sqrt{(n-1)/(nσ)}$ .最后，我们考虑了黎曼$n$ -流形$V$同构于$N\times [-1,1]$的情形，其中$N$是Rosenberg指数为非零的闭自旋流形。在这种情况下，我们表明，如果$\textrm{scal}(V)\geqσ>0$，那么$V$的边界组件之间的距离是最多$2π\sqrt{(n-1)/(nσ)}$。这最后一个常数是尖锐的论点由于格罗莫夫。

英文摘要：

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.

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