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Tubes in Complex Hyperbolic Manifolds

Ara Basmajian, Youngju Kim

Abstract

We prove a tubular neighborhood theorem for an embedded complex geodesic surface in a complex hyperbolic 2-manifold where the width of the tube depends only on the Euler characteristic of the embedded surface. We give an explicit estimate for this width. We supply two applications of the tubular neighborhood theorem, the first is a lower volume bound for such manifolds. The second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold. Finally, we prove a geometric combination theorem for two Fuchsian subgroups of PU(2,1). Using this combination theorem, we asymptotically bound (from above and below) the optimal width size of a tube about an embedded complex geodesic surface.

Tubes in Complex Hyperbolic Manifolds

Abstract

We prove a tubular neighborhood theorem for an embedded complex geodesic surface in a complex hyperbolic 2-manifold where the width of the tube depends only on the Euler characteristic of the embedded surface. We give an explicit estimate for this width. We supply two applications of the tubular neighborhood theorem, the first is a lower volume bound for such manifolds. The second is an upper bound on the first eigenvalue of the Laplacian in terms of the geometry of the manifold. Finally, we prove a geometric combination theorem for two Fuchsian subgroups of PU(2,1). Using this combination theorem, we asymptotically bound (from above and below) the optimal width size of a tube about an embedded complex geodesic surface.
Paper Structure (20 sections, 23 theorems, 111 equations, 5 figures, 1 table)

This paper contains 20 sections, 23 theorems, 111 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction and statement of results
  2. A little history
  3. Outline of proof
  4. Section plan and notation
  5. Complex hyperbolic geometry preliminaries
  6. Complex lines and orthogonal projection
  7. Normalizations
  8. Holonomy
  9. Holonomy and projection
  10. Areas, volumes, and wedges
  11. The Holonomy Angle Lemma and the Tubular Neighborhood Theorem
  12. Disjoint embedded surfaces
  13. The Eigenvalue problem
  14. Examples
  15. Example (No collar function for geodesics in a $3$-manifold)
  16. ...and 5 more sections

Key Result

Proposition 2.2

Let $L$ and $M$ be complex lines with polar vectors $\mathbf{n}, \mathbf{m}$ respectively, and set

Figures (5)

  • Figure 1: $d={\rho}(L_1, L_2) = {\rho}(\mathbf{0}, (x,0) )$
  • Figure 2: A wedge $W(s, \epsilon, \psi)$
  • Figure 3: A complex line $L$ and a bisector $B$
  • Figure 4: Bisector $B_q$
  • Figure 5: $\gamma (V_2) \subset U_1$

Theorems & Definitions (50)

  • Definition 2.1
  • Proposition 2.2: Goldman
  • Proposition 2.3
  • proof
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • proof
  • Proposition 2.7
  • ...and 40 more