The authors investigate minimal surfaces which solve a free boundary problem in a wedge \(W_\alpha\subset \mathbb{R}^3\) of opening angle \(\pi+\alpha\pi\) where \(-1<\alpha\leq 1\). Such minimal surfaces are given as conformal harmonic maps \(X= (X^1, X^2, X^3)\) from the half disc \(B= \{(u, v)\in \mathbb{R}^2\mid u^2+ v^2< 1,v>0\}\) into \(\mathbb{R}^3\) such that \(X\) maps the circular part of \(\partial B\) monotonically onto a Jordan arc joining the two open halfplanes which, together with the edge \(L\), form \(\partial W_\alpha\), whereas the straight part \(I\) of \(\partial B\) is mapped into \(\partial W_\alpha\). It is assumed that \(X\) is stationary for the Dirichlet integral implying that \(X\) meets \(\partial W_\alpha\) orthogonally along the free trace \(X(I)\) away from the edge \(L\). Choosing coordinates in such a way that \(L\) coincides with the \(x^3\)-axis and denoting by \(\pi_3\) the orthogonal projection onto the \((x^1, x^2)\)-plane, the authors introduce a further restriction on the minimal surface \(X^1\), namely \(f:= \pi_3\circ X\) should map the boundary segment \(I\) weakly monotonically into \(\pi_3(\partial W_\alpha)\). Then the two cases can occur: (i) \((f| I)^{-1}(0)\) is a single point, i.e., the free trace of \(X\) on \(\partial W_\alpha\) crosses the edge, (ii) \((f| I)^{-1}(0)\) is a proper interval, i.e., the free trace “creeps” along the edge. In the present paper mainly the case (i) is considered, (ii) has been dealt within the monograph “Minimal surfaces” (Vol. I: 1992; Zbl 0777.53012; Vol. II: 1992; Zbl 0777.53013) by U. Dierkes, S. Hildebrandt, A. Küster and O. Wohlrab.
Assuming that case (i) holds, \(f(0)= 0\), and the surface normal points into the half space \(x^3\geq 0\) near \(w= 0\) the main result of the paper is the following asymptotic development for the complex derivative \(X_w:={1\over 2}(X_u- iX_v)\) of \(X\) for \(w= u+ iv\) near \(w= 0:X_w= w^\alpha h_1j+ w^{-\alpha} h_2\overline j+ h_3 e_3\) where \(j:={1\over\sqrt 2} (e_1- ei_2)\), \(\overline j= {1\over\sqrt 2} (e_1+ ie_2)\) with \((e_1,e_2,e_3)\) the standard basis and functions \(h_1\), \(h_2\), \(h_3\) which extend holomorphically into the full disc \(u^2+ v^2< 1\). In the case \(\alpha= 1\) the formula can be strengthened to \(X_w= wH_1j+ w^3 H_2j+ H_3e_3\) with \(H_1\), \(H_2\), \(H_3\) holomorphic in the disc. The authors show furthermore that the total curvature of \(X\) is finite near the free trace.