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We study translation surfaces in the Euclidean 3-space $\mathbb{E}^3$ and the Gauss map $N$ with respect to the so-called Cheng-Yau operator $\square$. As a result, we prove that the only translation surfaces with Gauss map $N$ satisfying $\square N=AN$ for some $3\times3$ matrix $A$ are the flat ones. We also show that the only translation surfaces with Gauss map $N$ satisfying $\square N = A N$ for some nonzero $3\times3$ matrix $A$ are the cylindrical surfaces.
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참고문헌 (10)
참고문헌 신청
Luis J. Alias / 2006 / An extension of Takahashi theorem for the linearized operators of the higher order mean curvatures / Geom. Dedicata 121:113~127
C. Baikoussis / 1992 / On the Gauss map of ruled surfaces / Glasgow Math. J. 34(3):355~359
B.-Y. Chen / 1987 / Submanifolds with finite type Gauss map / Bull. Austral. Math. Soc. 35(2):161~186
S.Y. Cheng / 1977 / Hypersurfaces with constant scalar curvature / Math. Ann. 225(3):195~204
F. Dillen / 1990 / On the Gauss map of surfaces of revolution / Bull. Inst. Math. Acad. Sinica 18(3):239~246
C. Baikoussis / 1992 / On the Gauss map of ruled surfaces / Glasgow Math. J. 34(3):355~359
B.-Y. Chen / 1987 / Submanifolds with finite type Gauss map / Bull. Austral. Math. Soc. 35(2):161~186
S.Y. Cheng / 1977 / Hypersurfaces with constant scalar curvature / Math. Ann. 225(3):195~204
F. Dillen / 1990 / On the Gauss map of surfaces of revolution / Bull. Inst. Math. Acad. Sinica 18(3):239~246
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