Bottom spectrum and Llarull's theorem on complete noncompact manifolds

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.