Hasanis, Th.; Vlachos, Th. Surfaces of finite type with constant mean curvature. (English) Zbl 0828.53006 Kodai Math. J. 16, No. 2, 244-252 (1993). The authors prove the following results for finite type surfaces in the Euclidean 3-space \(E^3\). Theorem 1: A 3-type surface in \(E^3\) has nonconstant mean curvature. Theorem 2: Ordinary spheres, planes, catenoids and circular cylinders are the only surfaces of revolution with constant mean curvature in \(E^3\) which are of finite type. Reviewer: B.-Y.Chen (East Lansing) Cited in 5 Documents MSC: 53A05 Surfaces in Euclidean and related spaces 53B25 Local submanifolds 58J50 Spectral problems; spectral geometry; scattering theory on manifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) Keywords:surface of finite type; 2-type surface; 3-type surface; constant mean curvature PDFBibTeX XMLCite \textit{Th. Hasanis} and \textit{Th. Vlachos}, Kodai Math. J. 16, No. 2, 244--252 (1993; Zbl 0828.53006) Full Text: DOI References: [1] CHEN, B. -Y., Total Mean Curvature and Submanifolds of Finite Type, World Scientific publisher, 1984. · Zbl 0537.53049 [2] CHEN, B. -Y., Surfaces of finite type in Euclidean 3-space, Bull. Soc. Math. Belg Ser. B, 39 (1987), 243-254. · Zbl 0628.53011 [3] CHEN, B. -Y., 3-type surfaces in S3, Bull. Soc. Math. Belg. Ser. B, 42 (1990), 379-381 · Zbl 0732.53043 [4] CHEN, B. -Y. AND LI, S. J., 3-type hypersurfaces in a hypersphere, Bull. Soc Math. Belg. Ser. B, 43 (1991), 135-141. · Zbl 0752.53033 [5] HASANIS, T. AND KOUTROUFIOTIS, D., A property of complete minimal surfaces, Trans. Amer. Math. Soc. 281 (1984), 833-843 · Zbl 0538.53057 · doi:10.2307/2000089 [6] HASANIS, T. AND VLACHOS, T., Spherical 2-tye hypersurfaces, J. Geometr 40 (1991), 82-94. · Zbl 0726.53038 · doi:10.1007/BF01225875 [7] KAPOULEAS, N., Compact constant mean curvature surfaces in Euclidean three space, J. Diff. Geom. 33 (1991), 683-715. · Zbl 0727.53063 [8] PALAIS, R. S. AND TERNG, C. L., Critical Point Theory and Submanifold Geometry, Lecture Notes in Math. 1353, Springer-Verlag, 1988 · Zbl 0658.49001 · doi:10.1007/BFb0087442 [9] WENTE, H., Counter-example to the Hopf conjecture, Pac. J. Math. 121 (1986), 193-244 · Zbl 0586.53003 · doi:10.2140/pjm.1986.121.193 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.