Geodesic
Remarks on the regularity of quasislits
Lukas Schoug1 ; Atul Shekhar2 ; Fredrik Viklund3
1University of Cambridge, Faculty of Mathematics
2Université Lyon 1, Département Mathématiques
3KTH Royal Institute of Technology, Department of Mathematics
Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 355-370
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A quasislit is the image of a vertical line segment $[0, iy]$, $y > 0$, under a quasiconformal homeomorphism of the upper half-plane fixing $\infty$. Quasislits correspond precisely to curves generated by the Loewner equation with a driving function in the Lip-$\frac{1}{2}$ class. It is known that a quasislit is contained in a cone depending only on its Loewner driving function Lip-$\frac{1}{2}$ seminorm, $\sigma$. In this note we use the Loewner equation to give quantitative estimates on the opening angle of this cone in the maximal range $\sigma <4$. The estimate is shown to be sharp for small $\sigma$. As consequences, we derive explicit Hölder exponents for $\sigma < 4$ as well as estimates on winding rates. We also relate quantitatively the Lip-$\frac{1}{2}$ seminorm with the quasiconformal dilatation and discuss the optimal regularity of quasislits achievable through reparametrization.  
Keywords: Loewner differential equation, quasicircles, quasislits

Lukas Schoug  1   ; Atul Shekhar  2   ; Fredrik Viklund  3

1 University of Cambridge, Faculty of Mathematics
2 Université Lyon 1, Département Mathématiques
3 KTH Royal Institute of Technology, Department of Mathematics 
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Lukas Schoug; Atul Shekhar; Fredrik Viklund. Remarks on the regularity of quasislits. Annales Fennici Mathematici, Tome 46 (2021) no. 1, pp. 355-370. http://geodesic.mathdoc.fr/item/AFM_2021_46_1_a19/