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Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature

100%
López R., Demir E.
Open Mathematics
|
2014
|
tom 12
|
nr 9
1349-1361
EN
We classify all helicoidal non-degenerate surfaces in Minkowski space with constant mean curvature whose generating curve is a the graph of a polynomial or a Lorentzian circle. In the first case, we prove that the degree of the polynomial is 0 or 1 and that the surface is ruled. If the generating curve is a Lorentzian circle, we prove that the only possibility is that the axis is spacelike and the center of the circle lies on the axis.
2
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Polynomial translation surfaces of Weingarten types in Euclidean 3-space

88%
Yoon D.
Open Mathematics
|
2010
|
tom 8
|
nr 3
430-436
EN
In this paper, we classify polynomial translation surfaces in Euclidean 3-space satisfying the Jacobi condition with respect to the Gaussian curvature, the mean curvature and the second Gaussian curvature.
3
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Deforming metrics of foliations

63%
Rovenski V., Wolak R.
Open Mathematics
|
2013
|
tom 11
|
nr 6
1039-1055
EN
Let M be a Riemannian manifold equipped with two complementary orthogonal distributions D and D ⊥. We introduce the conformal flow of the metric restricted to D with the speed proportional to the divergence of the mean curvature vector H, and study the question: When the metrics converge to one for which D enjoys a given geometric property, e.g., is harmonic, or totally geodesic? Our main observation is that this flow is equivalent to the heat flow of the 1-form dual to H, provided the initial 1-form is D ⊥-closed. Assuming that D ⊥ is integrable with compact and orientable leaves, we use known long-time existence results for the heat flow to show that our flow has a solution converging to a metric for which H = 0; actually, under some topological assumptions we can prescribe the mean curvature H.
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