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# Helicoidal Minimal Surfaces in a Finsler Space of Randers Type

Published:2014-02-10
Printed: Dec 2014
• Rosângela Maria da Silva,
Instituto de Matemática e Estatística, IME-Universidade Federal de Goiás, 74001-970, Goiânia, GO, Brazil
• Keti Tenenblat,
Departamento de Matemática, Universidade de Brasília, 70904-970, Brasília, DF, Brazil
 Format: LaTeX MathJax PDF

## Abstract

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
 MSC Classifications: 53A10 - Minimal surfaces, surfaces with prescribed mean curvature [See also 49Q05, 49Q10, 53C42] 53B40 - Finsler spaces and generalizations (areal metrics)

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