Injectivity radius lower bound of convex sum of tame Riemannian metrics and applications to symplectic topology

Abstract

Motivated by the aspect of large-scale symplectic topology, we prove that for any pair $g_0,g_1$ of smooth complete Riemannian metrics of bounded curvature and of injectivity radius bounded away from zero, the convex sum $g_s:=(1-s)g_0+s{g}_1$ also has bounded curvature depending only on the curvature bounds ${\Vert R_{g_i}\Vert }_{C^0}$ of $g_0$ or $g_1$, and that the injectivity radii of $g_s$ have uniform lower bound depending only on the derivative bounds ${\Vert R_{g_i}\Vert }_{C^1}={\Vert R_{g_i}\Vert }_{C^0}+{\Vert D{R}_{g_i}\Vert }_{C^0}$. A main technical ingredient to establish the injectivity radius lower bound is an application of the quantitative inverse function theorem. Using these estimates, we prove that each quasi-isometry class of tame metrics is convex for all finite regularity class of $3\le r<\infty$. Using this Riemannian geometry result, we prove that the set of smooth $C^r$-tame almost complex structures inside the same quasi-isometry class associated to the symplectic form ω is contractible in strong $C^k$ topology for all $0\le k\le r$.