Minimal surfaces in a wedge. IV: Hölder estimates of the Gauss map and a Bernstein theorem
The authors study minimal graphs, i.e., solutions of the minimal surface equation \(z= \zeta(x,y)\) defined on a circular sector \[ \Omega_R=\{(\rho\cos \varphi, \rho\sin\varphi)\mid 0<\rho< R,\;0<\varphi<\pi+\alpha \pi\},\quad \alpha\in (-1,0)\cup (0,1), \] such that \(\zeta\) is continuous on \(\overline\Omega_R\) and fulfills the Neumann boundary condition along the straight line parts of \(\partial\Omega_R\). It is proved that there are constants \(c>0\), \(\theta\in (0,1)\) and \(\mu\in (0,1]\) such that the upwards pointing unit normal \(Y\) of any such minimal graph satisfies the estimate \(|(r\cos\varphi, r\sin\varphi)-(0,0,1)|\leq c(r,R)^\mu\) for all \(r\in (0,\theta R]\). It follows immediately that when such a minimal graph is defined on the infinite sector \(\Omega_\infty\) then it must necessarily be a horizontal plane. The latter result had already been obtained by \textit{C.-C. Lee} for \(\alpha\) between \(-1\) and \(0\) [Pac. J. Math. 177, No. 1, 103-107 (1997; Zbl 0868.53011)]. The proof is based on a representation of a minimal graph in conformal coordinates and uses ideas from earlier work of E. Heinz in order to estimate the map which accomplishes this change of coordinates. [See also Parts I--III in Calc. Var. Partial Differ. Equ. 5, No. 2, 99-115 (1997; Zbl 0920.49024); Arch. Math. 69, No. 2, 164-176 (1996; Zbl 0920.49025) and in J. Reine Angew. Math. 514, 71-101 (1999; Zbl 0940.49024)].
scientific article; zbMATH DE number 1271241