2024/06/27 by Daoqiang Liu, Liu, Daoqiang
Mathematics · #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2406.18997
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 \citeZh20. 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 \widehatA-area in \citeCZ24 to complete manifolds with compact boundary and then investigate width estimates of geodesic collar neighborhoods.