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Effective equidistribution of intersection points in hyperbolic manifolds

Tina Torkaman, Yongquan Zhang

TL;DR

The paper proves effective equidistribution of transverse intersection points between complementary-dimension geodesic submanifolds in finite-volume hyperbolic manifolds, including cusped cases, with explicit error terms that decay with volume growth and cusp geometry. It develops an intersection kernel on the frame bundle, couples smoothing techniques with homogeneous dynamics (geodesic flow) mixing results, and handles cuspidal excursions to achieve uniform bounds. The authors extend the results to joint equidistribution in the compact case via a configuration-space framework, and outline extensions and counterexamples that clarify the necessity of assumptions such as arithmeticity and maximality. The work advances quantitative understanding of geometric intersection patterns in hyperbolic geometry and provides tools potentially applicable to higher-rank settings and related configurations. Overall, the results yield robust, yet delicate, effective equidistribution statements with explicit dependence on cusp structure and geometric invariants.

Abstract

In this paper, we establish effective equidistribution of transverse intersection points between properly immersed totally geodesic submanifolds of complementary dimensions in a finite-volume hyperbolic manifold with respect to the hyperbolic volume measure, as the volume of the submanifolds tends to infinity.

Effective equidistribution of intersection points in hyperbolic manifolds

TL;DR

The paper proves effective equidistribution of transverse intersection points between complementary-dimension geodesic submanifolds in finite-volume hyperbolic manifolds, including cusped cases, with explicit error terms that decay with volume growth and cusp geometry. It develops an intersection kernel on the frame bundle, couples smoothing techniques with homogeneous dynamics (geodesic flow) mixing results, and handles cuspidal excursions to achieve uniform bounds. The authors extend the results to joint equidistribution in the compact case via a configuration-space framework, and outline extensions and counterexamples that clarify the necessity of assumptions such as arithmeticity and maximality. The work advances quantitative understanding of geometric intersection patterns in hyperbolic geometry and provides tools potentially applicable to higher-rank settings and related configurations. Overall, the results yield robust, yet delicate, effective equidistribution statements with explicit dependence on cusp structure and geometric invariants.

Abstract

In this paper, we establish effective equidistribution of transverse intersection points between properly immersed totally geodesic submanifolds of complementary dimensions in a finite-volume hyperbolic manifold with respect to the hyperbolic volume measure, as the volume of the submanifolds tends to infinity.

Paper Structure

This paper contains 42 sections, 23 theorems, 85 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Intersection of geodesics and hypersurfaces
  3. Generalization
  4. Counterexamples
  5. Joint equidistribution
  6. Discussion on related works
  7. Big $O$, small $o$
  8. Acknowledgment
  9. The intersection kernel
  10. Frames, submanifolds, and measures
  11. Intersection measure
  12. Intersection kernel
  13. Regularity and growth
  14. Effective equidistribution of geodesic submanifolds
  15. Equidistribution results
  16. ...and 27 more sections

Key Result

Theorem 1.1

Let $M$ be an arithmetic hyperbolic manifold of dimension $d \geq 3$. Let $\gamma_T$ be the sum of closed geodesics with length $\leq T$ and $\{S_n\}$ a sequence of distinct geodesic hypersurfaces in $M$. Then there exist positive constants $C, \epsilon_1,\epsilon_2,q$ depending only on the geometry where $\mathop{\mathrm{Sob}}\nolimits_{q}(f)$ is a degree $q$ Sobolev norm of $f$.

Figures (4)

  • Figure 1: $\alpha_n$, a closed geodesic that goes around the cusp $n$ times
  • Figure 2: How the map modifies a curve by removing an $R-$excursion
  • Figure 3: Plane for $\tau=\epsilon_d$ (corresponding to $\partial \mathcal{H}_{\epsilon_d}(\star)$) for $d=3$ tiled by fundamental domains of the lattice $\Lambda'$. The other endpoint of the geodesic excursion, $a_2$, lies in one of the shaded fundamental domains.
  • Figure 4: $\mathcal{N}_{\delta,\rho}(v_0)$ is the $\rho-$neighborhood of $B_{2\delta}^{d-1}(v_0)$.

Theorems & Definitions (63)

  • Theorem 1.1
  • Remark 1.2: On the Sobolev norm
  • Theorem 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Remark 1.6
  • Theorem 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 53 more