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

Voir la notice de l'article provenant de la source Cambridge University Press

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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