The Bj\"orling problem for non-minimal constant mean curvature surfaces

TL;DR

This paper extends the classical Bj"orling problem to non-minimal constant mean curvature surfaces, providing a method to explicitly construct solutions using loop group techniques and holomorphic data.

Contribution

It introduces a new approach to solve the Bj"orling problem for CMC surfaces via loop group formulation and elementary calculations, generalizing Schwarz's formula.

Findings

Explicit formula for holomorphic potential from Bj"orling data

Solution method using loop group and Iwasawa decomposition

Examples of CMC surfaces with special properties

Abstract

The classical Bj\"orling 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 Bj\"orling 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.