On the structure of certain Weingarten surfaces with boundary a circle
We call a surface \(M\) special Weingarten surface in \(\mathbb{R}^3\) if \(M\) satisfies the relation \(H=f (H^2-K)\), where \(H\) is the mean curvature of \(M\), \(K\) the Gaussian curvature of \(M\) and \(f\) a real smooth function such that \(4t (f(t))^2 <1\). The authors remark that they can apply maximum principle and Alexandrov reflection principle techniques to those surfaces as these were applied to constant mean curvature surfaces. They also look upon R. Bryant's construction of a global quadratic form \(Q\) on a surface \(M\) satisfying \(H= f(H^2-K)\), such that the zeros of \(Q\) are the umbilicial points of \(M\). So they characterize disk-type special Weingarten surfaces with boundary a round circle, extending some of their results for surfaces with constant mean curvature obtained in [\textit{F. Brito} and \textit{R. Sa Earp}, An. Acad. Bras. Cienc. 63, 5-19 (1991; Zbl 0803.53011)] and jointly with \textit{W. H. Meeks} and \textit{H. Rosenberg} in [Indiana Univ. Math. J. 40, 333-343 (1991; Zbl 0759.53003)].
scientific article; zbMATH DE number 1139495