Driving function of the vertical slit

Abstract

We consider the chordal Loewner equation and construct a family of vertical slits $\gamma^p\,(p>0)$. We give the exact expression of its driving function $\lambda$, and its H\"older exponent near $0$ in terms of $p$, which maps onto $(1/2, \infty)$ and has a natural connection with the known results. In addition, we extend the asymptotic behavior of the driving function $\lambda$ to the general case.