Geometric Description of Some Loewner Chains with Infinitely Many Slits

Abstract

We study the chordal Loewner equation associated with certain driving functions that produce infinitely many slits. Specifically, for a choice of a sequence of positive numbers (bn)n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(b_n)_{n\\ge 1}$$\\end{document} and points of the real line (kn)n≥1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(k_n)_{n\\ge 1}$$\\end{document}, we explicitily solve the Loewner PDE ∂f∂t(z,t)=-f′(z,t)∑n=1+∞2bnz-kn1-t\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\dfrac{\\partial f}{\\partial t}(z,t)=-f'(z,t)\\sum _{n=1}^{+\\infty }\\dfrac{2b_n}{z-k_n\\sqrt{1-t}} \\end{aligned}$$\\end{document}in H×[0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {H}\ imes [0,1)$$\\end{document}. Using techniques involving the harmonic measure, we analyze the geometric behaviour of its solutions, as t→1-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t\\rightarrow 1^-$$\\end{document}.