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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 Online first

Ha, Pham Hoang; Kawakami, Yu
A note on a unicity theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space
The classical result of Nevanlinna states that two nonconstant meromorphic functions on the complex plane having the same images for five distinct values must be identically equal to each other. In this paper, we give a similar uniqueness theorem for the Gauss maps of complete minimal surfaces in Euclidean four-space.

Keywords:minimal surface, Gauss map, unicity theorem
Categories:53A10, 30D35, 53C42

2. 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, spinors
Categories:53A10, 53C42

3. 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 space
Categories:53A10, 53B40

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