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 \({{\,\mathrm{scal}\,}}(X)\ge n(n-1)\) and there is a nonzero degree map into the sphere \(f:X\rightarrow S^n\) which is strictly area decreasing, then the distance between the support of \(\text {d}f\) and the boundary of X is at most \(\pi /n\). 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 \({{\,\mathrm{scal}\,}}(X)>\sigma >0\) 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 \(\pi \sqrt{(n-1)/(n\sigma )}\). 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 Rosenebrg index. In this case, we show that if \({{\,\mathrm{scal}\,}}(V)\ge \sigma >0\), then the distance between the boundary components of V is at most \(2\pi \sqrt{(n-1)/(n\sigma )}\). This last constant is sharp by an argument due to Gromov.

当拓扑信息由远离边界支持的束编码时，我们在具有边界的紧黎曼自旋流形上发展了指数理论。作为第一个应用，我们为边界为X的紧凑黎曼自旋n流形建立了“长颈原理” ，指出如果\（{{\，\ mathrm {scal} \，}}（X）\ ge n（ n-1）\），然后在球体（（f：X \ rightarrow S ^ n \）中有一个非零度图，严格地面积减小，然后\（\ text {d} f \ ），X的边界最多为\（\ pi / n \）。在旋转设置和严格减小面积的贴图中，这回答了Gromov最近提出的一个问题。作为第二应用中，我们考虑一个黎曼流形X通过除去所得ķ两两不相交嵌入ñ从闭合自旋-balls Ñ -manifold ÿ。我们证明如果\（{{\，\ mathrm {scal} \，}}（X）> \ sigma> 0 \）和Y满足以较高指数理论表示的特定条件，则测地线领的半径\（\ partial X \）的邻域最多为\（\ pi \ sqrt {（n-1）/（n \ sigma}} \）。最后，我们考虑黎曼的情况下ñ -manifold V微分形为\（N \ times [-1,1] \），其中N为封闭的自旋流形，其Rosenebrg指数不变。在这种情况下，我们证明如果\（{{\，\ mathrm {scal} \，}}（V）\ ge \ sigma> 0 \），则V的边界分量之间的距离最大为\（2 \ pi \ sqrt {（n-1）/（n \ sigma}} \）。由于格罗莫夫的说法，最后一个常数很明显。