A quantitative relative index theorem and Gromov's conjectures on positive scalar curvature

In this paper, we prove a quantitative relative index theorem. It provides a conceptual framework for studying some conjectures and open questions of Gromov on positive scalar curvature. More precisely, we prove a $\lambda$-Lipschitz rigidity theorem for (possibly incomplete) Riemannian metrics on spheres with certain types of subsets removed. This $\lambda$-Lipschitz rigidity theorem is asymptotically optimal. As a consequence, we obtain an asymptotically optimal $\lambda$-Lipschitz rigidity theorem for positive scalar curvature metrics on hemispheres. These give positive answers to the corresponding open questions raised by Gromov. As another application, we prove Gromov's $\square^{n-m}$ inequality on the bound of distances between opposite faces of spin manifolds with cube-like boundaries with a suboptimal constant. As immediate consequences, this implies Gromov's cube inequality on the bound of widths of Riemannian cubes and Gromov's conjecture on the bound of widths of Riemannian bands with suboptimal constants. Further geometric applications will be discussed in a forthcoming paper.