AbstractWe 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 $${{\,\mathrm{scal}\,}}(X)\ge n(n-1)$$
scal
(
X
)
≥
n
(
n
-
1
)
and there is a nonzero degree map into the sphere $$f:X\rightarrow S^n$$
f
:
X
→
S
n
which is strictly area decreasing, then the distance between the support of $$\text {d}f$$
d
f
and the boundary of X is at most $$\pi /n$$
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 $${{\,\mathrm{scal}\,}}(X)>\sigma >0$$
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 $$\partial X$$
∂
X
is at most $$\pi \sqrt{(n-1)/(n\sigma )}$$
π
(
n
-
1
)
/
(
n
σ
)
. Finally, we consider the case of a Riemannian n-manifold V diffeomorphic to $$N\times [-1,1]$$
N
×
[
-
1
,
1
]
, with N a closed spin manifold with nonvanishing Rosenebrg index. In this case, we show that if $${{\,\mathrm{scal}\,}}(V)\ge \sigma >0$$
scal
(
V
)
≥
σ
>
0
, then the distance between the boundary components of V is at most $$2\pi \sqrt{(n-1)/(n\sigma )}$$
2
π
(
n
-
1
)
/
(
n
σ
)
. This last constant is sharp by an argument due to Gromov.
Weighted Hardy inequality on Riemannian manifolds
