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The complex landslide flow and the method of integrable systems

Shimpei Kobayashi

TL;DR

The paper addresses how the complex landslide flow on Teichmüller-space pairs relates to an integrable-systems picture of harmonic maps into $\mathbb{H}^2$. It develops a correspondence via Labourie’s operator and CGC surfaces in $\mathbb{H}^3$, using harmonic maps, extended frames, and a spectral-parameter deformation within the loop-group framework. The main result shows that the holonomy of the complex landslide flow equals the holonomy of a family of flat connections evaluated at $\lambda = \sqrt{q}$, with the flow $q \mapsto P_q(h,h^{\star})$ holomorphic due to the holomorphic dependence of the extended frame on $\lambda$. This provides a unified, integrable-systems perspective on complex landslides and their relation to complex earthquakes, with potential applications in Teichmüller theory and CGC-surface geometry.

Abstract

We investigate a connection between the complex landslide flow, defined on a pair of Teichmüller spaces, and the integrable system approach to harmonic maps into a symmetric space. We will prove that the holonomy of the complex landslide flow can be derived from the holonomy of the family of flat connections determined by a harmonic map into the hyperbolic two-space.

The complex landslide flow and the method of integrable systems

TL;DR

The paper addresses how the complex landslide flow on Teichmüller-space pairs relates to an integrable-systems picture of harmonic maps into . It develops a correspondence via Labourie’s operator and CGC surfaces in , using harmonic maps, extended frames, and a spectral-parameter deformation within the loop-group framework. The main result shows that the holonomy of the complex landslide flow equals the holonomy of a family of flat connections evaluated at , with the flow holomorphic due to the holomorphic dependence of the extended frame on . This provides a unified, integrable-systems perspective on complex landslides and their relation to complex earthquakes, with potential applications in Teichmüller theory and CGC-surface geometry.

Abstract

We investigate a connection between the complex landslide flow, defined on a pair of Teichmüller spaces, and the integrable system approach to harmonic maps into a symmetric space. We will prove that the holonomy of the complex landslide flow can be derived from the holonomy of the family of flat connections determined by a harmonic map into the hyperbolic two-space.
Paper Structure (7 sections, 10 theorems, 58 equations)

This paper contains 7 sections, 10 theorems, 58 equations.

Table of Contents

  1. Introduction and the main result
  2. Preliminaries
  3. Surfaces in the hyperbolic three-space
  4. Natural projections, two Gauss maps and a Ruh-Vilms type theorem
  5. The extended frames and CGC surfaces
  6. The complex landslides flow and CGC surfaces
  7. Proof of the main theorem

Key Result

Theorem 1.1

Let $P_q(h, h^{\star})\; (q \in {\overline \mathbb D}^{\times}, (h, h^{\star}) \in \mathcal{T}\times \mathcal{T})$ be the complex landslide flow. Then there exists a family of flat connections $\nabla^{\lambda}$ of a harmonic map into $\mathbb H^2$ corresponding to $(h, h^{\star})$ such that the hol

Theorems & Definitions (21)

  • Theorem 1.1
  • Corollary 1.1
  • Remark 1.1
  • Definition 2.1
  • Theorem 2.1
  • Theorem 2.2: Theorem B (1) in IK:CGCH3
  • Theorem 2.3: Theorem 2.1 in IK:CGCH3
  • Remark 2.1
  • Lemma 2.1
  • proof
  • ...and 11 more