Conformal welding of independent quantum disks for all $\gamma\in [0,\sqrt{2}]$ using the Beltrami equation.

Problem's Description

SLE$(\kappa)$ describes a curve growing in time describing the interface of many statistical models that fall within conformal field theory. The research problem here is to construct a family of loops using the Beltrami equation that might be related to the SLE-loop.

In the works [1] and [2] this was achieved for small enough $\kappa\leq \kappa_{0}$.

The SLE-loop has already being constructed in various works: [3], that constructed an SLE loop measure using a two-sided whole-plane SLE but also proved a criterion for uniqueness up to a constant to unify the various constructions; Sheffield and Werner CLE measure [4], Kemppainen and Werner CLE's Mobius-invariance [5], Miller and Schoug CLE for $\kappa\in (8/3, 8) $ [6], Fields and Lawler "SLE loops rooted at an interior point" , Benoist, S., Dubédat, J.: "Building $SLE_\kappa$ loop measures for $\kappa < 4$". Also, there was the construction in [8][9] by Morris, Holden, and Sun, who showed in ([theorem 1.3])[8] that the construction of SLE loops based on the quantum zipper satisfies Zhan's uniqueness.

The difference of the above with the approach in the works [1] and [2] is that here the construction starts from proving the existence of the welding using the standard theory of Beltrami equations (see [11] and [10]). So one is still left with the task of proving a relation to SLE. However, it provides a complementary approach that is more analytic in nature and thus can provide a new perspective on many of the achievements in imaginary geometry.

The research problem is to extend the existence and uniqueness of the conformal welding over the entire range $\kappa\in [0,4]$, and thus including the critical case which will likely be the hardest.

1. ArticleIs an originInverse of the Gaussian multiplicative chaos: Lehto welding of Independent Quantum disks

   • Ilia Binder,
   • Tomas Kojar

   arXiv preprint arXiv:2311.18163, 2023

2. ArticleIs an originConformal welding of independent Gaussian multiplicative chaos measures

   • Antti Kupiainen,
   • Michael McAuley,
   • Eero Saksman

   arXiv preprint arXiv:2305.18062, 2023

3. Article SLE loop measures

   • Dapeng Zhan

   Probability Theory and Related Fields 179, 345-406, 2021

4. Article Conformal loop ensembles: the Markovian characterization and the loop-soup construction

   • Scott Sheffield,
   • Wendelin Werner

   Annals of Mathematics, 1827-1917, 2012

5. Article The nested simple conformal loop ensembles in the Riemann sphere

   • Antti Kemppainen,
   • Wendelin Werner

   Probability Theory and Related Fields 165, 835-866, 2016

6. Article Existence and uniqueness of the conformally covariant volume measure on conformal loop ensembles

   • Jason Miller,
   • Lukas Schoug

   arXiv preprint arXiv:2201.01748, 2022

7. Article Conformal weldings of random surfaces: SLE and the quantum gravity zipper

   • Scott Sheffield

   Annals of Probability 44, 3474-3545, 2016

8. Article Integrability of SLE via conformal welding of random surfaces

   • Morris Ang,
   • Nina Holden,
   • Xin Sun

   arXiv preprint arXiv:2104.09477, 2021

9. Article Conformal welding of quantum disks

   • Morris Ang,
   • Nina Holden,
   • Xin Sun

   arXiv preprint arXiv:2009.08389, 2020

10. Book Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (PMS-48)

    • Kari Astala,
    • Tadeusz Iwaniec,
    • Gaven Martin

    year of publication: 2008

11. Article Random conformal weldings

    • Kari Astala,
    • Peter Jones,
    • Antti Kupiainen,
    • Eero Saksman

    year of publication: 2011