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

Sebastian Boldt, Batu Güneysu, Stefano Pigola

TL;DR

The paper derives a pointwise lower bound for the normal injectivity radius of embedded hypersurfaces in arbitrary Riemannian manifolds using a local Blaschke ball rolling framework and two-sided radial angle comparisons. It then translates these geometric controls into practical consequences: a computable lower bound for the graphing radius, the construction of metrics of bounded geometry on manifolds with boundary, and a compatibility result between geometric and topological notions of orientation in metric measure spaces. A key technical achievement is the ability to convexify the boundary via conformal changes without losing uniform tubular neighborhoods, enabling RCD-type analysis on manifolds with convex boundaries. Overall, the work bridges classical differential geometry with metric-measure geometry, with implications for geometric analysis, flows, and L^2-cohomology on noncompact spaces with boundary.

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.

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

TL;DR

The paper derives a pointwise lower bound for the normal injectivity radius of embedded hypersurfaces in arbitrary Riemannian manifolds using a local Blaschke ball rolling framework and two-sided radial angle comparisons. It then translates these geometric controls into practical consequences: a computable lower bound for the graphing radius, the construction of metrics of bounded geometry on manifolds with boundary, and a compatibility result between geometric and topological notions of orientation in metric measure spaces. A key technical achievement is the ability to convexify the boundary via conformal changes without losing uniform tubular neighborhoods, enabling RCD-type analysis on manifolds with convex boundaries. Overall, the work bridges classical differential geometry with metric-measure geometry, with implications for geometric analysis, flows, and L^2-cohomology on noncompact spaces with boundary.

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.

Paper Structure

This paper contains 8 sections, 44 theorems, 174 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Local radial angle comparison
  3. A local Blaschke ball rolling theorem and the normal injectivity radius
  4. Intersections of hypersurfaces and convex bodies
  5. Hypersurfaces as graphs
  6. Submanifolds of bounded geometry
  7. Convexification of the boundary
  8. Orientations of smooth metric measure spaces with boundary

Key Result

Theorem A

Let $(M,g)$ be a Riemannian manifold, $\Sigma\subseteq M$ an embedded hypersurface and $x\in\Sigma$. Let $c,\lambda \geq 0$ and $s > 0$ such that Then

Figures (5)

  • Figure 1: A connected hypersurface with arbitrarily small normal radius around flat points
  • Figure 2: The situation of Proposition \ref{['prop:distance-trajectories-of-X']}
  • Figure 3: A gradient line of ${\mathrm{d}_p}{|\Sigma_r}$ that is tangent to $\partial \mathsf{B}_r(x)$.
  • Figure 4: Proof of Theorem \ref{['thm:local-Blaschke']}.
  • Figure 5: The left-hand figure illustrates Theorem \ref{['thm:blaschke-outer-ball-rolling-theorem']}. The right-hand figure illustrates the application of Theorem \ref{['thm:blaschke-intersection-ball-theorem']} as in case 3 in the proof of Theorem \ref{['thm:normal-radius-est-sigma-intersection-convex-domain']}.

Theorems & Definitions (98)

  • Theorem A: Klingenberg's lemma for embedded hypersurfaces, Theorem \ref{['thm:Klingenberg-hypersurface']}
  • Theorem B: Local two-sided inner Blaschke ball rolling theorem, Theorem \ref{['thm:local-Blaschke']}
  • Theorem : Blaschke Inclusion Theorem Blaschke
  • Theorem C: Theorem \ref{['thm:normal-radius-est-sigma-intersection-convex-domain']}
  • Corollary D: Corollary \ref{['cor:lambda-convex-body-in-Sigma-tubular-neighbourhood']}
  • Theorem E: Theorem \ref{['thm:graphing-radius-estimate']}
  • Theorem F: Corollary \ref{['cor:existence-metric-bounded-geometry-boundary']}
  • Theorem G: Theorem \ref{['thm:bounded-geometry-convex-boundary']}
  • Definition 2.1
  • Remark 2.2
  • ...and 88 more