Bottom spectrum and Llarull's theorem on complete noncompact manifolds
Daoqiang Liu
TL;DR
The paper extends Llarull's theorem to complete noncompact spin manifolds and to manifolds with boundary by relating lower bounds on the scalar curvature to the bottom spectrum of the Laplacian via deformed Dirac (Callias) operators. Central to the approach are GL-pairs, Callias operators $\mathcal{B}_f=\mathcal{D}+f\sigma$, and spectral flow and relative index techniques, coupled with area-decreasing maps $\Phi: M\to \mathbf{S}^m$ of nonzero degree that are locally constant at infinity. The main result provides a quantitative bound: for any $c>\frac{m-1}{4m}$, $\inf_M \operatorname{scal}_{g_M} < -\frac{1}{c}\lambda_1(M,g_M)$ under $\operatorname{scal}_{g_M}\ge m(m-1)$ on $\operatorname{supp}(\mathrm{d}\Phi)$, with a boundary analogue yielding $\inf_M \operatorname{scal}_{g_M} < 0$ under mild boundary conditions. The method yields a non-approximation statement and a boundary version dropping strict boundary positivity, and is illustrated via the established framework of spectral flow and relative indices in the Callias setting. Overall, the work connects scalar curvature obstructions to spectral data on noncompact spaces, reinforcing rigidity phenomena in geometric analysis.
Abstract
In this paper, we prove an extension of the noncompact version of Llarull's theorem due to Zhang and Li-Su-Wang-Zhang, giving an upper bound for the infimum of scalar curvature in terms of the bottom spectrum of the Laplacian. Moreover, we extend the theorem to manifolds with boundary, relaxing the strict positivity condition on the scalar curvature near the boundary that was required by Liu-Liu. Our approach is based on deformed Dirac operators.