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Rigidity of proper holomorphic maps between balls with Hölder boundary regularity

Jaan Amla Srimurthy, Kyle Huang, Jinwoo Park, Aleksander Skenderi, Ruo Wen, Andrew Zimmer

TL;DR

The work proves a rigidity theorem for proper holomorphic maps $f: \\mathbb B^m \to \\mathbb B^M$ with Hölder boundary regularity exponent $\\alpha>\\tfrac{1}{2}$ under a strong symmetry assumption (the projection of the symmetry group $\\mathsf{G}_f$ to $\\mathsf{Aut}(\\mathbb B^m)$ is Zariski dense). The strategy combines dynamics on the ball boundary, the Fundamental Theorem of Affine Geometry, and analytic envelope techniques: first show $f$ maps affine lines to affine lines along loxodromic directions, extend this to all lines using Zariski density, deduce rationality, and finally invoke prior results to normalize $f$ by automorphisms to $(z,0)$. The argument relies on a hyperboloid model, contraction rates of loxodromics, and real-analyticity properties of envelopes, bridging complex hyperbolic geometry with affine-geometry rigidity. The result connects to conjectures about representations in complex hyperbolic space, providing a concrete path to rationality and a canonical normal form under symmetry and Hölder regularity. Overall, the paper advances a symmetry-driven rigidity paradigm for holomorphic maps between balls, with potential applications to complex hyperbolic lattices and geometric analysis of holomorphic maps.

Abstract

In this paper, we prove a rigidity result for proper holomorphic maps between unit balls that have many symmetries and which extend to Hölder continuous maps on the boundary, with Hölder exponent strictly greater than 1/2.

Rigidity of proper holomorphic maps between balls with Hölder boundary regularity

TL;DR

The work proves a rigidity theorem for proper holomorphic maps with Hölder boundary regularity exponent under a strong symmetry assumption (the projection of the symmetry group to is Zariski dense). The strategy combines dynamics on the ball boundary, the Fundamental Theorem of Affine Geometry, and analytic envelope techniques: first show maps affine lines to affine lines along loxodromic directions, extend this to all lines using Zariski density, deduce rationality, and finally invoke prior results to normalize by automorphisms to . The argument relies on a hyperboloid model, contraction rates of loxodromics, and real-analyticity properties of envelopes, bridging complex hyperbolic geometry with affine-geometry rigidity. The result connects to conjectures about representations in complex hyperbolic space, providing a concrete path to rationality and a canonical normal form under symmetry and Hölder regularity. Overall, the paper advances a symmetry-driven rigidity paradigm for holomorphic maps between balls, with potential applications to complex hyperbolic lattices and geometric analysis of holomorphic maps.

Abstract

In this paper, we prove a rigidity result for proper holomorphic maps between unit balls that have many symmetries and which extend to Hölder continuous maps on the boundary, with Hölder exponent strictly greater than 1/2.
Paper Structure (16 sections, 24 theorems, 120 equations)

This paper contains 16 sections, 24 theorems, 120 equations.

Table of Contents

  1. Introduction
  2. Outline of the proof
  3. Background
  4. Possibly ambiguous notation
  5. The automorphism group of the unit ball
  6. The limit set and North-South Dynamics
  7. Zariski Density
  8. Real analytic sets
  9. Asymptotic behavior of automorphisms
  10. The Image of Lines Through Fixed Points
  11. Affine lines intersecting the limit set
  12. Analytic sets
  13. Analytic envelopes
  14. The hyperboloid model
  15. Proof of Theorem \ref{['thm:analytic envelopes']}
  16. ...and 1 more sections

Key Result

Theorem 1.3

If $2 \leq m < M$ and $f: \mathop{\mathrm{\mathbb{B}}}\nolimits^m \rightarrow \mathop{\mathrm{\mathbb{B}}}\nolimits^M$ is a proper holomorphic map where then there exist $\phi_1 \in \mathop{\mathrm{\mathsf{Aut}}}\nolimits(\mathop{\mathrm{\mathbb{B}}}\nolimits^m)$ and $\phi_2 \in \mathop{\mathrm{\mathsf{Aut}}}\nolimits(\mathop{\mathrm{\mathbb{B}}}\nolimits^M)$ such that for all $z \in \mathop{\ma

Theorems & Definitions (48)

  • Example 1.1
  • Conjecture 1.2
  • Theorem 1.3
  • Conjecture 1.4
  • Conjecture 1.5
  • Theorem 1.6: The Fundamental Theorem of Affine Geometry, see e.g. McCallum
  • Theorem 1.7: see Theorem \ref{['thm:loxo lines are mapped to loxo lines']} below
  • Theorem 2.1: see e.g. Abate1989
  • Theorem 2.2
  • proof
  • ...and 38 more