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

Motivated by the aspect of large-scale symplectic topology, we prove that for any pair $g_0,g_1$ of complete Riemannian metrics of bounded curvature and \emph{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 $\mathrm{\parallel }{R}_{g_i}{\mathrm{\parallel }}_{C^0}$ of $g_0$ or $g_1$, and that the injectivity radii of $g_s$ have uniform lower bound depending only on the derivative bounds $\mathrm{\parallel }{R}_{g_i}{\mathrm{\parallel }}_{C^1}=\mathrm{\parallel }{R}_{g_i}{\mathrm{\parallel }}_{C^0}+\mathrm{\parallel }D{R}_{g_i}{\mathrm{\parallel }}_{C^0}$. 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 and so contractible in strong $C^r$ topology for all finite regularity class of $3\le r<\mathrm{\infty }$. Using this Riemannian geometry result, we prove that the set of $C^3$-tame almost complex structures inside the same quasi-isometry class associated to the symplectic form $\omega$ is contractible.