Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 41, 2016, 681-691

ON TANGENTIAL SLIT SOLUTION OF THE LOEWNER EQUATION

Ka-Sing Lau and Hai-Hua Wu

The Chinese University of Hong Kong, Department of Mathematics
Shatin, N.T., Hong Kong; kslau 'at' math.cuhk.edu.hk

The Chinese University of Hong Kong, Department of Mathematics, Hong Kong
and Hunan University, College of Mathematics and Econometrics
Changsha, Hunan 410082, P.R. China; hunaniwa 'at' 163.com

Abstract. For a non-tangential slit γ(t), the behavior of the driving function λ(t) near zero in the Loewner equation is well understood; for tangential slit, the situation is less clear. In this paper, we investigate the tangential slit Γ^p, p >0, where Γ is a circular arc tangent at 0; Γ^p has order p+1/p near zero. Our main result is to give the exact expression of λ(t), and its Hölder exponent near 0 in terms of p, which has a natural connection with the known results. We also extend this to a general type of tangential slits, and give an estimation of the growth of λ(t) near 0.

2010 Mathematics Subject Classification: Primary 30C45, 30C20; Secondary 30C62, 30C30.

Key words: Loewner equation, hull, half-plane capacity, driving function, trace.

Reference to this article: K.-S. Lau and H.-H. Wu: On tangential slit solution of the Loewner equation. Ann. Acad. Sci. Fenn. Math. 41 (2016), 681-691.

Full document as PDF file

doi:10.5186/aasfm.2016.4142