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Asymptotically geodesic surfaces

Fernando Al Assal, Ben Lowe

TL;DR

The paper investigates sequences of surfaces in hyperbolic and negatively curved 3-manifolds whose principal curvatures tend to zero. It proves a density dichotomy: in finite-volume hyperbolic 3-manifolds, asymptotically geodesic surfaces are dense in the 2-plane Grassmann bundle $\mathrm{Gr} M$, while in geometrically finite infinite-volume manifolds such sequences cannot exist. It further establishes rigidity phenomena: if such sequences persist under appropriate metric identifications, the ambient metric is forced to be hyperbolic (or the manifold to have constant curvature in the totally umbilic setting). In higher dimensions, accumulation sets of asymptotically Fuchsian surfaces can be finite unions of totally geodesic submanifolds, realized concretely via arithmetic constructions, and the paper also constructs examples of dense but not asymptotically geodesic minimal surfaces. Overall, the results connect dynamical rigidity for geodesic flows with geometric constraints on low-curvature surfaces, yielding new rigidity and nonexistence phenomena across dimensions.

Abstract

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the 2-plane Grassmann bundle of M. When M has infinite volume and is geometrically finite, we show such sequences do not exist. As an application of the former, we obtain partial answers to the question of whether a negatively curved Riemannian 3-manifold that contains a sequence of asymptotically totally geodesic or totally umbilic surfaces must be hyperbolic. Finally, we give examples to show that if the dimension of M is greater than 3, the possible limiting behavior of asymptotically geodesic surfaces is less constrained than for totally geodesic surfaces.

Asymptotically geodesic surfaces

TL;DR

The paper investigates sequences of surfaces in hyperbolic and negatively curved 3-manifolds whose principal curvatures tend to zero. It proves a density dichotomy: in finite-volume hyperbolic 3-manifolds, asymptotically geodesic surfaces are dense in the 2-plane Grassmann bundle , while in geometrically finite infinite-volume manifolds such sequences cannot exist. It further establishes rigidity phenomena: if such sequences persist under appropriate metric identifications, the ambient metric is forced to be hyperbolic (or the manifold to have constant curvature in the totally umbilic setting). In higher dimensions, accumulation sets of asymptotically Fuchsian surfaces can be finite unions of totally geodesic submanifolds, realized concretely via arithmetic constructions, and the paper also constructs examples of dense but not asymptotically geodesic minimal surfaces. Overall, the results connect dynamical rigidity for geodesic flows with geometric constraints on low-curvature surfaces, yielding new rigidity and nonexistence phenomena across dimensions.

Abstract

A sequence of distinct closed surfaces in a hyperbolic 3-manifold M is asymptotically geodesic if their principal curvatures tend uniformly to zero. When M has finite volume, we show such sequences are always asymptotically dense in the 2-plane Grassmann bundle of M. When M has infinite volume and is geometrically finite, we show such sequences do not exist. As an application of the former, we obtain partial answers to the question of whether a negatively curved Riemannian 3-manifold that contains a sequence of asymptotically totally geodesic or totally umbilic surfaces must be hyperbolic. Finally, we give examples to show that if the dimension of M is greater than 3, the possible limiting behavior of asymptotically geodesic surfaces is less constrained than for totally geodesic surfaces.

Paper Structure

This paper contains 23 sections, 27 theorems, 7 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Density of asymptotically geodesic surfaces in finite volume
  3. Hausdorff limits in higher dimensions
  4. Nonexistence of asymptotically geodesic surfaces in infinite volume
  5. Rigidity of asymptotically geodesic surfaces in negative curvature
  6. Rigidity of totally umbilic surfaces in negative curvature
  7. Beyond asymptotically Fuchsian
  8. Related work and context
  9. Outline
  10. Acknowledgements
  11. Proof of Theorem \ref{['contra']}
  12. When $M$ has finitely many finite area totally geodesic surfaces
  13. When $M$ has infinitely many finite area totally geodesic surfaces
  14. Nonexistence of asymptotically geodesic surfaces in infinite volume
  15. Possible accumulation sets in higher dimensions
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $M$ be a finite volume hyperbolic 3-manifold. Let $B$ be an open set. Then, there is $\epsilon=\epsilon(B)>0$ such that all but finitely many of the $(1+\epsilon)$-quasifuchsian minimal surfaces of $M$ meet $B$. Moreover, the ones which do not are totally geodesic.

Figures (1)

  • Figure 1: The scrawly curve represents the surfaces $S(\epsilon_j,R_j)$, which become dense in $N_1\cup N_2$ as $j\to\infty$.

Theorems & Definitions (43)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Remark 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Theorem 1.10
  • ...and 33 more