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Nearly geodesic surfaces are filling

Xiaolong Hans Han

Abstract

Let $M$ be a closed hyperbolic $3$-manifold. A homotopy class $[S]$ of surfaces in $M$ is filling if any representative cuts $M$ into components contractible in $M$. We prove that there exist $ε_0, g_0>0$ such that every homotopy class of $(1+ε)$-quasi-Fuchsian surfaces with $0<ε\leq ε_0$ or totally geodesic surfaces of genus $\geq g_0$ in $M$ is filling. As a corollary, except for at most finitely many totally geodesic surfaces, embedded incompressible quasi-Fuchsian surfaces in $M$ have constants bounded below by $1+ε_0$. This also gives a gap theorem for embedded minimal surfaces. Each of these surfaces separates any pair of distinct points at the sphere of infinity. Crucial tools include the rigidity results of Mozes-Shah, Ratner, and Shah. This work is inspired by a question of Wu and Xue whether random geodesics on random hyperbolic surfaces are filling.

Nearly geodesic surfaces are filling

Abstract

Let be a closed hyperbolic -manifold. A homotopy class of surfaces in is filling if any representative cuts into components contractible in . We prove that there exist such that every homotopy class of -quasi-Fuchsian surfaces with or totally geodesic surfaces of genus in is filling. As a corollary, except for at most finitely many totally geodesic surfaces, embedded incompressible quasi-Fuchsian surfaces in have constants bounded below by . This also gives a gap theorem for embedded minimal surfaces. Each of these surfaces separates any pair of distinct points at the sphere of infinity. Crucial tools include the rigidity results of Mozes-Shah, Ratner, and Shah. This work is inspired by a question of Wu and Xue whether random geodesics on random hyperbolic surfaces are filling.

Paper Structure

This paper contains 18 sections, 34 theorems, 41 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. The motivations from hyperbolic geometry and random geodesics
  3. The motivations from the rigidity of unipotent flows
  4. Preliminaries
  5. Quasi-Fuchsian surfaces and minimal surfaces
  6. A natural metric on the Grassmann bundles
  7. Fundamental domains and fillingness
  8. Strongly filling surfaces in hyperbolic $3$-manifolds
  9. Some lemmas on intersections and fillingness
  10. A filling totally geodesic surface is strongly filling
  11. All but finitely many totally geodesic hypersurfaces are strongly filling
  12. Nearly geodesic surfaces are strongly filling
  13. The dichotomy of embedded surfaces in a hyperbolic $3$-manifold
  14. Least area non-Fuchsian surfaces fail to minimize intersection numbers
  15. Cubulations and one-orbit of hyperplanes
  16. ...and 3 more sections

Key Result

Theorem 1.1

Let $M$ be a closed hyperbolic $3$-manifold. There exist $\mathfrak{g}_0, \epsilon_0 >0$ such that excluding finitely many classes of totally geodesic surfaces of genus $<\mathfrak{g}_0$, all $(1+\epsilon_0)$-quasi-Fuchsian surfaces are strongly filling.

Figures (6)

  • Figure 1: The filling nature of nearly geodesic surfaces in a closed hyperbolic $3$-manifold (each dot represents a $(1+\epsilon_0)$-quasi-Fuchsian surface). The constant $C$ is the universal constant coming from Seppi's curvature estimate saMinimalDiscsHyperbolicSpace.
  • Figure 2: The weak-$^*$ limit is supported on two components. The Hausdorff limit connects the two components by two geodesics which are neither closed nor dense and forms a connected subset of the line bundle over $S$.
  • Figure 3: The least area $S$ has redundant intersections with some geodesic $\gamma$
  • Figure 4: Fundamental domains of the six closed flat orientable $3$-manifolds. Unlabeled faces are glued by translations.
  • Figure 5: Filling surfaces in flat manifolds
  • ...and 1 more figures

Theorems & Definitions (67)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.6
  • Theorem 1.7: kmImmersingAlmostGeodesicbwBoundaryCriterionCubulation
  • Corollary 1.8
  • Corollary 1.9
  • Theorem 1.10
  • ...and 57 more