Geodesic
Helicoidal Minimal Surfaces in a Finsler Space of Randers Type
Canadian mathematical bulletin, Tome 57 (2014) no. 4, pp. 765-779

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DOI

We consider the Finsler space $\left( {{\overline{M}}^{3}},\,\overline{F} \right)$ 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 ${{\overline{M}}^{3}}$ . We prove that the helicoid is a minimal surface in ${{\overline{M}}^{3}}$ only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $\left( {{\overline{M}}^{3}},\,\overline{F} \right)$ , the only minimal surfaces in the Bonnet family with fixed axis $O{{\overline{x}}^{3}}$ are the catenoids and the helicoids.
DOI : 10.4153/CMB-2013-047-7
Mots-clés : 53A10, 53B40, minimal surfaces, helicoidal surfaces, Finsler space, Randers space 
Silva, Rosângela Maria da; Tenenblat, Keti. Helicoidal Minimal Surfaces in a Finsler Space of Randers Type. Canadian mathematical bulletin, Tome 57 (2014) no. 4, pp. 765-779. doi: 10.4153/CMB-2013-047-7
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     author = {Silva, Ros\^angela Maria da and Tenenblat, Keti},
     title = {Helicoidal {Minimal} {Surfaces} in a {Finsler} {Space} of {Randers} {Type}},
     journal = {Canadian mathematical bulletin},
     pages = {765--779},
     year = {2014},
     volume = {57},
     number = {4},
     doi = {10.4153/CMB-2013-047-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-2013-047-7/}
}
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