On distance estimates for complete manifolds with lower scalar curvature bounds

In this paper, we focus on the distance estimate problem on complete manifolds with compact boundary and with lower scalar curvature bounds. On these manifolds, relative to a background manifold with nonnegative curvature operator, we introduce a definition of the relative index of relative Gromov-Lawson pairs via a deformed Dirac operator trick in \cite{Zh20}. We prove that the relative index coincides with the index of associated Callias operators of the relative Gromov-Lawson pairs. As applications, we prove a short neck inequality with uniformly positive scalar curvature and the corresponding quantitative shielding result with nonnegative scalar curvature. Moreover, we generalize the concept of relative $\widehat{A}$-area in \cite{CZ24} to complete manifolds with compact boundary and then investigate width estimates of geodesic collar neighborhoods.