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It is well known that the helicoids are the only ruled minimal surfaces in $\RRR ^3$. The similar characterization for ruled minimal surfaces can be given in many other 3-dimensional homogeneous spaces. In this note we consider the product space $M\times\RRR$ for a 2-dimensional manifold $M$ and prove that $M\times\RRR$ has a nontrivial minimal surface ruled by horizontal geodesics only when $M$ has a Clairaut parametrization. Moreover such minimal surface is the trace of the longitude rotating in $M$ while translating vertically in constant speed in the direction of $\RRR$.
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참고문헌 (7)
참고문헌 신청
M. Do Carmo / 1983 / Rotation hypersurfaces in spaces of constant curvature / Trans. Amer. Math. Soc 277(2):685~709
Y. W. Kim / 2009 / Helicoids in S2 ×R and H2 ×R / Pacific J. Math 242(2):281~297
H. B. Lawson / 1970 / Complete minimal surfaces in S3 / Ann. Math 92:335~374
B. O’neill / 1997 / Elementary Differential Geometry / Academic Press
H. Shin / 2013 / Ruled minimal surfaces in the three dimensional Heisenberg group / Pacific J. Math 261(2):477~496
Y. W. Kim / 2009 / Helicoids in S2 ×R and H2 ×R / Pacific J. Math 242(2):281~297
H. B. Lawson / 1970 / Complete minimal surfaces in S3 / Ann. Math 92:335~374
B. O’neill / 1997 / Elementary Differential Geometry / Academic Press
H. Shin / 2013 / Ruled minimal surfaces in the three dimensional Heisenberg group / Pacific J. Math 261(2):477~496
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