Abstract
Let \(\mathbb {D}\subset \mathbb {C}\) be the unit disk and let \(\gamma_{1},\gamma _{2}:[0,T]\to\overline{\mathbb {D}}\setminus\{0\}\) be parametrizations of two slits \(\Gamma_{1}:=\gamma(0,T], \Gamma_{2}:=\gamma_{2}(0,T]\) such that \(\Gamma_{1}\) and \(\Gamma_{2}\) are disjoint.
Let \(g_{t}\) be the unique normalized conformal mapping from \(\mathbb {D}\setminus(\gamma_{1}[0,t]\cup\gamma_{2}[0,t])\) onto \(\mathbb {D}\) with \(g_{t}(0)=0\), \(g'_{t}(0)>0\). Furthermore, for \(k=1,2\), denote by \(h_{k;t}\) the unique normalized conformal mapping from \(\mathbb {D}\setminus\gamma_{k}[0,t]\) onto \(\mathbb {D}\) with \(h_{k;t}(0)=0\), \({h'_{k;t}(0)}>0\).
Loewner’s famous theorem (1923) can be stated in the following way: The function \(t\mapsto h_{k;t}\) is differentiable at \(t_{0}\) if and only if \(t\mapsto\log(h_{k;t}'(0))\) is differentiable at \(t_{0}\).
In this paper we compare the differentiability of \(t\mapsto h_{k;t}\) with that of \(t\mapsto g_{t}\). We show that the situation is more complicated in the case \(t_{0}=0\) with \(\gamma_{1}(0)=\gamma_{2}(0)\).
Furthermore, we also look at this problem in the case of a multiply connected domain with its corresponding Komatu–Loewner equation.
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References
Bauer, R. O. and Friedrich, R. M., On radial stochastic Loewner evolution in multiply connected domains, J. Funct. Anal. 237 (2006), 565–588.
Böhm, C. and Lauf, W., A Komatu–Loewner equation for multiple slits, Comput. Methods Funct. Theory 14 (2014), 639–663.
Böhm, C. and Schleißinger, S., Constant coefficients in the radial Komatu–Loewner equation for multiple slits, Math. Z. 279 (2015), 321–332.
Conway, J. B., Functions of One Complex Variable. II, Graduate Texts in Mathematics 159, Springer, New York, 1995.
Komatu, Y., Untersuchungen über konforme Abbildung von zweifach zusammenhängenden Gebieten, Proc. Phys.-Math. Soc. Jpn. 25 (1943), 1–42.
Komatu, Y., On conformal slit mapping of multiply-connected domains, Proc. Jpn. Acad. 26 (1950), 26–31.
Lawler, G. F., Conformally Invariant Processes in the Plane, Mathematical Surveys and Monographs 114, American Mathematical Society, Providence, RI, 2005.
Lawler, G. F., Schramm, O. and Werner, W., Values of Brownian intersection exponents. I. Half-plane exponents, Acta Math. 187 (2001), 237–273.
Lind, J., Marshall, D. E. and Rohde, S., Collisions and spirals of Loewner traces, Duke Math. J. 154 (2010), 527–573.
Löwner, K., Untersuchungen über schlichte konforme Abbildungen des Einheitskreises. I, Math. Ann. 89 (1923), 103–121.
Pommerenke, C., Boundary Behaviour of Conformal Maps, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 299, Springer, Berlin, 1992.
Renggli, H., An inequality for logarithmic capacities, Pacific J. Math. 11 (1961), 313–314.
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The second author was partially supported by the ERC grant “HEVO–Holomorphic Evolution Equations” no. 277691.
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Böhm, C., Schleißinger, S. The Loewner equation for multiple slits, multiply connected domains and branch points. Ark Mat 54, 339–370 (2016). https://doi.org/10.1007/s11512-016-0231-9
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DOI: https://doi.org/10.1007/s11512-016-0231-9