Summary
For conformai maps g on the closed unit disc \( \overline B \) we shall estimate \( \left\| g \right\|\,_{C^{2 + a} \,(\bar B)} \) by the relevant geometric data from above. Here we bound the modulus of their derivatives \( \left| {g'(w)} \right| > 0,w \in \,\overline B \) quantitatively from below. Only the maximum principle and Minding’s formula for the geodesic curvature are necessary in the proof. For instance, our estimates suffice to construct conformai maps of the class \( {C^{{2 + \alpha }}}(\overline B ) \) approximating C 2+α-domains by real-analytic Jordan-domains.
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References
W. Blaschke: Vorlesungen über Differentialgeometrie I. Springer, Berlin, 1945
R. Courant: Dirichlet’s principle, conformai mapping, and minimal surfaces. Springer, Berlin, Reprint 1977
U. Dierkes, S. Hildebrandt, et al.: Minimal surfaces II. Grundlehren der math. Wiss. 296. Springer, Berlin, 1992
H. Grauert: Funktionentheorie I. Vorlesungsausarbeitung am Mathematischen Institut der Universität Göttingen, Wintersemester 1964/65
C. Pommerenke: Univalent functions. Vandenhoek und Ruprecht, Göttingen, 1975
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© 2003 Springer-Verlag Berlin Heidelberg
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Sauvigny, F. (2003). Global C 2+α-Estimates for Conformai Maps. In: Hildebrandt, S., Karcher, H. (eds) Geometric Analysis and Nonlinear Partial Differential Equations. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55627-2_7
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DOI: https://doi.org/10.1007/978-3-642-55627-2_7
Publisher Name: Springer, Berlin, Heidelberg
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