Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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SINGULAR MINIMAL TRANSLATION GRAPHS IN EUCLIDEAN SPACES

  • Received : 2019.12.18
  • Accepted : 2020.09.21
  • Published : 2021.01.01
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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.

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References

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