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Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

Pinsard Morel Lucas

TL;DR

The paper studies the local topology of the group of symplectomorphisms on the real tori by constructing a gradient flow from a moment map associated to a (hyper)Kähler structure. A DeTurck-type gauge is applied to convert the invariant gradient flow into a strictly parabolic evolution, yielding local existence, uniqueness, and regularity results for the flow in both the 2- and 4-dimensional torus cases. These local well-posedness results imply the local contractibility of the corresponding symplectomorphism groups. The work extends Rollin’s ideas and integrates hyperkähler geometry to handle the $ ext{T}^4$ case, providing a parallel framework for analyzing the topology of symplectomorphism groups on even-dimensional tori.

Abstract

Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its use in the study of the Ricci flow), showing the local existence and regularity for the solutions of this flow and hence showing that the group of symplectomorphisms of the real four-dimensional torus is locally contractible. This work follows the ideas introduced by Yann Rollin in [3], even though the moment map picture comes from different considerations.

Local Existence Of The Symplectic Gradient Flow On The Hyperkähler Four-dimensional Flat Torus

TL;DR

The paper studies the local topology of the group of symplectomorphisms on the real tori by constructing a gradient flow from a moment map associated to a (hyper)Kähler structure. A DeTurck-type gauge is applied to convert the invariant gradient flow into a strictly parabolic evolution, yielding local existence, uniqueness, and regularity results for the flow in both the 2- and 4-dimensional torus cases. These local well-posedness results imply the local contractibility of the corresponding symplectomorphism groups. The work extends Rollin’s ideas and integrates hyperkähler geometry to handle the case, providing a parallel framework for analyzing the topology of symplectomorphism groups on even-dimensional tori.

Abstract

Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its use in the study of the Ricci flow), showing the local existence and regularity for the solutions of this flow and hence showing that the group of symplectomorphisms of the real four-dimensional torus is locally contractible. This work follows the ideas introduced by Yann Rollin in [3], even though the moment map picture comes from different considerations.

Paper Structure

This paper contains 15 sections, 60 equations.

Table of Contents

  1. Introduction
  2. Motivations
  3. Notations
  4. Main Results
  5. Case of $\mathbb{T}^2$
  6. Setting and properties
  7. Moment map and gradient flow
  8. DeTurck Trick
  9. Theorem for $\mathbb{T}^2$
  10. Case of $\mathbb{T}^4$
  11. Setting and properties
  12. Moment map and gradient flow
  13. DeTurck Trick
  14. Theorem for $\mathbb{T}^4$
  15. Conclusion and perspectives

Theorems & Definitions (17)

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