Lower bounds on the normal injectivity radius of hypersurfaces and bounded geometries on manifolds with boundary

2025/04/03 by Sebastian Boldt, Boldt, Sebastian, Batu Güneysu +3
Mathematics · #53B20 (Primary) #53B25 #53C18 #53C20 #53C40 #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Holomorphic and Operator Theory

paper · pdf · doi:10.48550/arxiv.2504.02687

openalex publication_date 2025/04/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We prove for the first time a pointwise lower estimate of the normal injectivity radius of an embedded hypersurface in an arbitrary Riemannian manifold. Main applications include: (i) a pointwise lower estimate of the graphing radius of a properly embedded hypersurface; (ii) the construction of metrics of bounded geometry on arbitrary manifolds with boundary; (iii) the equivalence of the classical (topological) notion of orientation with that of the geometric notion (in the sense of metric measure spaces) on arbitrary Riemannian manifolds with boundary. In addition, we prove that every manifold with boundary admits a metric with bounded geometry such that the boundary becomes convex. This result strengthens the justification of a recent notion of orientation on finite dimensional RCD spaces.

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