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Fueter equations and the search for a higher dimensional Hamiltonian Floer theory I: analytical foundations and compactness

L. Asselle, R. Brilleslijper

Abstract

We study a Floer-theoretic approach to harmonic maps from the two-torus into non-flat Kähler manifolds. Building on the complex-regularized polysymplectic (CRPS) formalism of [BF24], which provides a Hamiltonian description of harmonic maps for which the associated equations are elliptic, we analyze the compactness of the associated moduli spaces of Fueter maps. For compact quotients $Q$ of complex hyperbolic space, we exploit the structure of the Biquard-Gauduchon hyperkähler metric to prove relative compactness under suitable smallness assumptions on the Hamiltonian. In the flat case, we establish the necessary quantitative $L^\infty$-estimates and outline a perturbative strategy for the non-flat setting.

Fueter equations and the search for a higher dimensional Hamiltonian Floer theory I: analytical foundations and compactness

Abstract

We study a Floer-theoretic approach to harmonic maps from the two-torus into non-flat Kähler manifolds. Building on the complex-regularized polysymplectic (CRPS) formalism of [BF24], which provides a Hamiltonian description of harmonic maps for which the associated equations are elliptic, we analyze the compactness of the associated moduli spaces of Fueter maps. For compact quotients of complex hyperbolic space, we exploit the structure of the Biquard-Gauduchon hyperkähler metric to prove relative compactness under suitable smallness assumptions on the Hamiltonian. In the flat case, we establish the necessary quantitative -estimates and outline a perturbative strategy for the non-flat setting.
Paper Structure (7 sections, 16 theorems, 123 equations)

This paper contains 7 sections, 16 theorems, 123 equations.

Table of Contents

  1. Introduction
  2. CRPS geometry
  3. Hyperkähler metrics on cotangent bundles
  4. Compactness for Fueter curves
  5. Bubbling analysis
  6. Singularities
  7. A note on the a priori $L^\infty$-estimates

Key Result

Theorem 1.2

Let $Q$ be a compact quotient of complex hyperbolic space by a group of isometries and let $\mathbb D^*_{\delta_0}Q\subseteq T^*Q$ be a hyperkähler neighborhood of the zero section. Then, there exists $0<\delta_*<\delta_0$ such that the moduli space $\mathcal{M}$ of Fueter maps defined in sec:hyperk

Theorems & Definitions (34)

  • Remark 1.1
  • Theorem 1.2
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Proposition 2.4
  • proof
  • Theorem 2.5: Theorem 7.2 from CRPS
  • Theorem 3.1: feix2001hyperkahler
  • Remark 3.2
  • ...and 24 more