Cheng-Yau operator and Gauss map of Translation Surfaces
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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.
Abstract
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.
참고문헌 (10)
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