Geodesic
On the conformal metric of annulus in the n-dimensional Euclidean
Dalʹnevostočnyj matematičeskij žurnal, Tome 18 (2018) no. 2, pp. 233-241

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It is shown by the methods of symmetrization that the geodesic with respect to the conformal metric of annulus in the Euclidean space is located into a two-dimensional sector. As a consequence, the geodesic is established in the case of points located on symmetric sphere of the annulus. Exact lower bounds are proved for the conformal metric of the annulus. A distortion theorem for quasi-regular mappings is given. 
@article{DVMG_2018_18_2_a11,
     author = {E. G. Prilepkina and A. S. Afanaseva-Grigoreva},
     title = {On the conformal metric of annulus  in the n-dimensional {Euclidean}},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {233--241},
     publisher = {mathdoc},
     volume = {18},
     number = {2},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2018_18_2_a11/}
}
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E. G. Prilepkina; A. S. Afanaseva-Grigoreva. On the conformal metric of annulus  in the n-dimensional Euclidean. Dalʹnevostočnyj matematičeskij žurnal, Tome 18 (2018) no. 2, pp. 233-241. http://geodesic.mathdoc.fr/item/DVMG_2018_18_2_a11/