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On the Bergman Kernel of hyperbolic Riemann surfaces

Jingzhou Sun

TL;DR

This work addresses the long-standing problem of an explicit Bergman kernel formula on hyperbolic Riemann surfaces by reducing to tractable local models (disk, cylinder, cusp) and then summing over the Fuchsian group. The authors develop a three-step strategy: (i) exact cylinder and cusp analyses capturing holonomy and geodesic data, (ii) an enlargement-and-limit process to relate cusps to cylinders, and (iii) a global summation over Γ to pass from local to global kernels. The main contribution is an exact diagonal formula for rho_k(p) in terms of a geodesic-loop sum with holonomy weighting, plus sharp off-diagonal decay and a max/min phenomenon tied to the interplay between short loops and holonomies. This Selberg-trace-like framework exposes how global geometry and holonomy dictate Bergman kernel behavior on finite-topology and infinite-volume hyperbolic surfaces, with potential implications for systolic geometry and quantum phenomena near cusps.

Abstract

We prove an exact formula for the Bergman kernel function of hyperbolic Riemann surfaces either of finite topology or of positive injectivity radius. The formula involves summation over all geodesic loops based at a point, which has a striking analogy with the Selberg trace formula. As an application, we prove a result about the maximum and minimum of the Bergman kernel function. We also prove an estimate of the off-diagonal Bergman kernel.

On the Bergman Kernel of hyperbolic Riemann surfaces

TL;DR

This work addresses the long-standing problem of an explicit Bergman kernel formula on hyperbolic Riemann surfaces by reducing to tractable local models (disk, cylinder, cusp) and then summing over the Fuchsian group. The authors develop a three-step strategy: (i) exact cylinder and cusp analyses capturing holonomy and geodesic data, (ii) an enlargement-and-limit process to relate cusps to cylinders, and (iii) a global summation over Γ to pass from local to global kernels. The main contribution is an exact diagonal formula for rho_k(p) in terms of a geodesic-loop sum with holonomy weighting, plus sharp off-diagonal decay and a max/min phenomenon tied to the interplay between short loops and holonomies. This Selberg-trace-like framework exposes how global geometry and holonomy dictate Bergman kernel behavior on finite-topology and infinite-volume hyperbolic surfaces, with potential implications for systolic geometry and quantum phenomena near cusps.

Abstract

We prove an exact formula for the Bergman kernel function of hyperbolic Riemann surfaces either of finite topology or of positive injectivity radius. The formula involves summation over all geodesic loops based at a point, which has a striking analogy with the Selberg trace formula. As an application, we prove a result about the maximum and minimum of the Bergman kernel function. We also prove an estimate of the off-diagonal Bergman kernel.

Paper Structure

This paper contains 25 sections, 35 theorems, 249 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background and preliminaries
  3. Notation and conventions
  4. Disk model
  5. Estimates for the Bergman kernel using geodesic balls
  6. Hyperbolic cylinder model
  7. Setting up
  8. Main result
  9. Exponential Decay
  10. Exact formula
  11. Proof of the main theorem for hyperbolic cylinder
  12. Twisted hyperbolic cylinder
  13. Holonomy along a geodesic loop
  14. Proof of Theorem \ref{['thm-main-cylinder']}
  15. Cusp part
  16. ...and 10 more sections

Key Result

Theorem 1.1

Suppose that $(X,\omega)$ is a Riemann surface equipped with a complete hyperbolic metric. Let $(L,h)$ be a Hermitian holomorphic line bundle over $X$, whose curvature is $-i\omega$. We further assume that $(X,\omega)$ either has finite topology(i.e., genus of $X$ is finite), or the injectivity radi

Figures (1)

  • Figure 1: Quadrilateral used in the discussion of the twisted hyperbolic cylinder.

Theorems & Definitions (61)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Conjecture 1.5
  • Proposition 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Theorem 3.3
  • ...and 51 more