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http://dx.doi.org/10.4134/JKMS.j190851

SINGULAR MINIMAL TRANSLATION GRAPHS IN EUCLIDEAN SPACES  

Aydin, Muhittin Evren (Department of Mathematics Firat University)
Erdur, Ayla (Department of Mathematics Tekirdag Namik Kemal University)
Ergut, Mahmut (Department of Mathematics Tekirdag Namik Kemal University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.1, 2021 , pp. 109-122 More about this Journal
Abstract
In this paper, we consider the problem of finding the hypersurface Mn in the Euclidean (n + 1)-space ℝn+1 that satisfies an equation of mean curvature type, called singular minimal hypersurface equation. Such an equation physically characterizes the surfaces in the upper half-space ℝ+3 (u) with lowest gravity center, for a fixed unit vector u ∈ ℝ3. We first state that a singular minimal cylinder Mn in ℝn+1 is either a hyperplane or a α-catenary cylinder. It is also shown that this result remains true when Mn is a translation hypersurface and u is a horizantal vector. As a further application, we prove that a singular minimal translation graph in ℝ3 of the form z = f(x) + g(y + cx), c ∈ ℝ - {0}, with respect to a certain horizantal vector u is either a plane or a α-catenary cylinder.
Keywords
Singular minimal hypersurface; translation hypersurface; cylinder; translation graph; ${\alpha}$-catenary;
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