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Tetrahedra in complex hyperbolic space and Hilbert spaces with Pick kernels

Richard Rochberg

Abstract

We study of the relation between the geometry of sets in complex hyperbolic space and Hilbert spaces with complete Pick kernels. We focus on the geometry associated with assembling sets into larger sets and of assembling Hilbert spaces into larger spaces. Model questions include describing the possible triangular faces of a tetrahedron in hyperbolic space and describing the three dimensional subspaces of four dimensional Hilbert spaces with Pick kernels. Our novel technical tool is a complex analog of the cosine of a vertex angle.

Tetrahedra in complex hyperbolic space and Hilbert spaces with Pick kernels

Abstract

We study of the relation between the geometry of sets in complex hyperbolic space and Hilbert spaces with complete Pick kernels. We focus on the geometry associated with assembling sets into larger sets and of assembling Hilbert spaces into larger spaces. Model questions include describing the possible triangular faces of a tetrahedron in hyperbolic space and describing the three dimensional subspaces of four dimensional Hilbert spaces with Pick kernels. Our novel technical tool is a complex analog of the cosine of a vertex angle.

Paper Structure

This paper contains 36 sections, 41 theorems, 71 equations.

Table of Contents

  1. Introduction and Summary
  2. Background
  3. Hyperbolic Geometry
  4. Hilbert Spaces with Reproducing Kernels
  5. Rescaling
  6. Assuming Irreducibility
  7. The CPP
  8. Invariants
  9. Matrix Notation
  10. The $\mathcal{CPP}$ and Point Sets in $\mathbb{CH}^{n}$
  11. Assembly and Coherence
  12. Equivalence of the Two Questions
  13. Geometry and $Kos$
  14. Evaluating $Kos$
  15. $Kos$ and other Invariants
  16. ...and 21 more sections

Key Result

Theorem 3.1

Given a three dimensional $H\in\mathcal{RK}$ the following are equivalent: Furthermore $X$ sits inside a complex geodesic if and only if $\det \operatorname*{KOS}(H,1)=0,$ or equivalently $\left\vert \operatorname*{kos}\nolimits_{1}(2,3)\right\vert =1,$ or the $b$ coordinate of $\Gamma$ in (3) equals $0.$ Conversely, given data $S$ and $\Gamma$'s defined from $S$ using (ij

Theorems & Definitions (66)

  • Definition 2.1
  • Theorem 3.1: Br AM Ro
  • proof
  • Theorem 3.2
  • Corollary 3.3
  • Proposition 3.4
  • proof
  • Proposition 4.1
  • proof
  • Proposition 4.2
  • ...and 56 more