Minimal surfaces are two-dimensional objects with zero mean curvature at every point. In this paper, we will demonstrate how this condition can be expressed using the coefficients of the first and second fundamental forms. Furthermore, we will prove that each point on such a surface possesses a neighborhood with a harmonic isothermal parametrization, where the partial derivatives with respect to both parameters are orthogonal and of equal length, while the coordinate functions are harmonic.
Utilizing these coordinates, we will establish the existence of a unique minimal surface containing a given continuous differentiable curve, along which the expected surface normal field is prescribed. Such a surface is referred to as a solution to the Björling problem.
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