Ruled submanifolds of finite type

Ruled submanifolds of finite type
HTML articles powered by AMS MathViewer

Proc. Amer. Math. Soc. 114 (1992), 795-798

DOI: https://doi.org/10.1090/S0002-9939-1992-1072333-5

PDF | Request permission

We show that a ruled submanifold of finite type in a Euclidean space is a cylinder on a curve of finite type or a generalized helicoid.

References

Günter Aumann, Die Minimalhyperregelflächen, Manuscripta Math. 34 (1981), no. 2-3, 293–304 (German, with English summary). MR 620454, DOI 10.1007/BF01165542
J. M. Barbosa, M. Dajczer, and L. P. Jorge, Minimal ruled submanifolds in spaces of constant curvature, Indiana Univ. Math. J. 33 (1984), no. 4, 531–547. MR 749313, DOI 10.1512/iumj.1984.33.33028
D. E. Blair and J. R. Vanstone, A generalization of the helicoid, Minimal submanifolds and geodesics (Proc. Japan-United States Sem., Tokyo, 1977) North-Holland, Amsterdam-New York, 1979, pp. 13–16. MR 574249
Bang-Yen Chen, Total mean curvature and submanifolds of finite type, Series in Pure Mathematics, vol. 1, World Scientific Publishing Co., Singapore, 1984. MR 749575, DOI 10.1142/0065
Bang-Yen Chen, Franki Dillen, Leopold Verstraelen, and Luc Vrancken, Ruled surfaces of finite type, Bull. Austral. Math. Soc. 42 (1990), no. 3, 447–453. MR 1083281, DOI 10.1017/S0004972700028616
H. Blaine Lawson Jr., Complete minimal surfaces in $S^{3}$, Ann. of Math. (2) 92 (1970), 335–374. MR 270280, DOI 10.2307/1970625
Yu. G. Lumiste, Die $n$-dimensionalen Minimalflächen mit einer $(n-1)$-dimensionalen asymptotischen Richtung in jedem Punkte, Tartu Riikl. Ül. Toimetised 62 (1958), 117–141 (Russian, with Estonian and German summaries). MR 0110057
James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556

Minimal generalized ruled surfaces in the Euclidean space