Communications in Analysis and Geometry

Volume 18 (2010)

Number 1

The Björling problem for non-minimal constant mean curvature surfaces

Pages: 171 – 194

DOI: http://dx.doi.org/10.4310/CAG.2010.v18.n1.a7

Authors

David Brander (Department of Mathematics, Technical University of Denmark)

Josef F. Dorfmeister (TU München, Zentrum Mathematik, Garching, Germany)

Abstract

The classical Björling problem is to find the minimal surface containing a given real analytic curve with tangent planes prescribed along the curve. We consider the generalization of this problem to non-minimal constant mean curvature (CMC) surfaces, and show that it can be solved via the loop group formulation for such surfaces. The main result gives a way to compute the holomorphic potential for the solution directly from the Björling data, using only elementary differentiation, integration and holomorphic extensions of real analytic functions. Combined with an Iwasawa decomposition of the loop group, this gives the solution, in analogue to Schwarz’s formula for the minimal case. Some preliminary examples of applications to the construction of CMC surfaces with special properties are given.

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