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In this paper, we consider the problem of finding the hypersurface $M^{n}$ in the Euclidean $\left( n+1\right)$-space $\mathbb{R}^{n+1}$ that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper halfspace $\mathbb{R}_{+}^{3}\left( \mathbf{u} \right) $ with lowest gravity center, for a fixed unit vector $\mathbf{u}\in \mathbb{R}^{3}$. We first state that a singular minimal cylinder $M^{n}$ in $\mathbb{R}^{n+1}$ is either a hyperplane or a $\alpha $-catenary cylinder. It is also shown that this result remains true when $M^{n}$ is a translation hypersurface and $\mathbf{u}$ is a horizantal vector. As a further application, we prove that a singular minimal translation graph in $\mathbb{R }^{3}$ of the form $z=f(x)+g(y+cx),$ $c\in \mathbb{R-\{}0\},$ with respect to a certain horizantal vector $\mathbf{u}$ is either a plane or a $\alpha $- catenary cylinder.
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참고문헌 (32)
참고문헌 신청
M. E. Aydin / 2015 / Translation hypersurfaces and Tzitzeica translation hypersurfaces of the Euclidean space / Proc. Rom. Acad. Ser. A Math. Phys. Tech. Sci. Inf. Sci 16(4):477~483
B.-Y. Chen / 1973 / Geometry of Submanifolds / Marcel Dekker, Inc
B.-Y. Chen / 2011 / On some geometric properties of h-homogeneous production functions in microeconomics / Kragujevac J. Math 35(3):343~357
B.-Y. Chen / 2011 / Pseudo-Riemannian Geometry, δ-Invariants and Applications / World Scientific Publishing Co. Pte. Ltd
U. Dierkes / 1993 / A Bernstein result for energy minimizing hypersurfaces / Calculus of Variations and Partial Differential Equations 1(1):37~54
B.-Y. Chen / 1973 / Geometry of Submanifolds / Marcel Dekker, Inc
B.-Y. Chen / 2011 / On some geometric properties of h-homogeneous production functions in microeconomics / Kragujevac J. Math 35(3):343~357
B.-Y. Chen / 2011 / Pseudo-Riemannian Geometry, δ-Invariants and Applications / World Scientific Publishing Co. Pte. Ltd
U. Dierkes / 1993 / A Bernstein result for energy minimizing hypersurfaces / Calculus of Variations and Partial Differential Equations 1(1):37~54
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