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Search: MSC category 53A10 ( Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] )

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1. CMB 2015 (vol 59 pp. 50)

Dorfmeister, Josef F.; Inoguchi, Jun-ichi; Kobayashi, Shimpei
 On the Bernstein Problem in the Three-dimensional Heisenberg Group In this note we present a simple alternative proof for the Bernstein problem in the three-dimensional Heisenberg group $\operatorname{Nil}_3$ by using the loop group technique. We clarify the geometric meaning of the two-parameter ambiguity of entire minimal graphs with prescribed Abresch-Rosenberg differential. Keywords:Bernstein problem, minimal graphs, Heisenberg group, loop groups, spinorsCategories:53A10, 53C42

2. CMB 2014 (vol 57 pp. 765)

da Silva, Rosângela Maria; Tenenblat, Keti
 Helicoidal Minimal Surfaces in a Finsler Space of Randers Type We consider the Finsler space $(\bar{M}^3, \bar{F})$ obtained by perturbing the Euclidean metric of $\mathbb{R}^3$ by a rotation. It is the open region of $\mathbb{R}^3$ bounded by a cylinder with a Randers metric. Using the Busemann-Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in $\bar{M}^3$. We prove that the helicoid is a minimal surface in $\bar{M}^3$, only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $(\bar{M}^3, \bar{F})$, the only minimal surfaces in the Bonnet family, with fixed axis $O\bar{x}^3$, are the catenoids and the helicoids. Keywords:minimal surfaces, helicoidal surfaces, Finsler space, Randers spaceCategories:53A10, 53B40
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