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

Jaeyoung Choi; Yong-Geun Oh

doi:10.1016/j.aim.2025.110443

paper

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

arXiv:2401.09777 · doi:10.1016/j.aim.2025.110443

Abstract

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 $∥ R_{g_{i}} ∥_{C^{0}}$ of $g_{0}$ or $g_{1}$ , and that the injectivity radii of $g_{s}$ have uniform lower bound depending only on the derivative bounds $∥ R_{g_{i}} ∥_{C^{1}} = ∥ R_{g_{i}} ∥_{C^{0}} + ∥ D R_{g_{i}} ∥_{C^{0}}$ . 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 $3 \leq r < \infty$ . 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 $ω$ is contractible.

33 pages, comments welcome!; v2) 35 pages, main contractibility results improved in strong $C^{3}$ topology, title sightly changed, introduction partially rewritten and rearranged; v3) 38 pages, final version in press. Advances in Math

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

Abstract

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