| Title:
|
Translation surfaces of finite type in ${\rm Sol}_{3}$ (English) |
| Author:
|
Senoussi, Bendehiba |
| Author:
|
Al-Zoubi, Hassan |
| Language:
|
English |
| Journal:
|
Commentationes Mathematicae Universitatis Carolinae |
| ISSN:
|
0010-2628 (print) |
| ISSN:
|
1213-7243 (online) |
| Volume:
|
61 |
| Issue:
|
2 |
| Year:
|
2020 |
| Pages:
|
237-256 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In the homogeneous space Sol$_{3}$, a translation surface is parametrized by $r(s,t)=\gamma _{1}(s)\ast \gamma _{2}(t)$, where $\gamma _{1}$ and $\gamma _{2}$ are curves contained in coordinate planes. In this article, we study translation invariant surfaces in ${\rm Sol}_{3}$, which has finite type immersion. (English) |
| Keyword:
|
Laplacian operator |
| Keyword:
|
homogeneous space |
| Keyword:
|
invariant surface |
| Keyword:
|
surfaces of coordinate finite type |
| MSC:
|
53B25 |
| MSC:
|
53C30 |
| idZBL:
|
Zbl 07286003 |
| idMR:
|
MR4143707 |
| DOI:
|
10.14712/1213-7243.2020.018 |
| . |
| Date available:
|
2020-10-13T13:15:12Z |
| Last updated:
|
2022-07-04 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148283 |
| . |
| Reference:
|
[1] Al-Zoubi H., Stamatakis S., Al-Mashaleh W., Awadallah M.: Translation surfaces of coordinate finite type.Indian J. Math. 59 (2017), no. 2, 227–241. MR 3700538 |
| Reference:
|
[2] Bekkar M., Senoussi B.: Translation surfaces in the $3$-dimensional space satisfying $\Delta ^{III}r_{i}=\mu _{i}r_{i}$.J. Geom. 103 (2012), no. 3, 367–374. MR 3017050, 10.1007/s00022-012-0136-0 |
| Reference:
|
[3] Chen B.-Y.: Total Mean Curvature and Submanifolds of Finite Type.Series in Pure Mathematics, 1, World Scientific Publishing, Singapore, 1984. Zbl 0537.53049, MR 0749575 |
| Reference:
|
[4] Dillen F., Verstraelen L., Zafindratafa G.: A generalization of the translation surfaces of Scherk.Differential Geometry in honor of Radu Rosca. K. U. L. (1991), 107–109. |
| Reference:
|
[5] Inoguchi J., López R., Munteanu M.-I.: Minimal translation surfaces in the Heisenberg group $ Nil_{3}$.Geom. Dedicata 161 (2012), 221–231. MR 2994039, 10.1007/s10711-012-9702-8 |
| Reference:
|
[6] López R., Munteanu M. I.: On the geometry of constant angle surfaces in $ Sol_{3}$.Kyushu J. Math. 65 (2011), no. 2, 237–249. MR 2977760, 10.2206/kyushujm.65.237 |
| Reference:
|
[7] López R., Munteanu M. I.: Minimal translation surfaces in $ Sol_{3}$.J. Math. Soc. Japan. 64 (2012), no. 3, 985–1003. MR 2965436, 10.2969/jmsj/06430985 |
| Reference:
|
[8] Scott P.: The geometries of $3$-manifolds.Bull. London Math. Soc. 15 (1983), no. 5, 401–487. Zbl 0662.57001, MR 0705527, 10.1112/blms/15.5.401 |
| Reference:
|
[9] Takahashi T.: Minimal immersions of Riemannian manifolds.J. Math. Soc. Japan 18 (1966), 380–385. MR 0198393, 10.2969/jmsj/01840380 |
| Reference:
|
[10] Troyanov M.: L'horizon de $ SOL$.Exposition. Math. 16 (1998), no. 5, 441–479. MR 1656902 |
| Reference:
|
[11] Yoon D. W.: Coordinate finite type invariant surfaces in $ Sol$ spaces.Bull. Iranian Math. Soc. 43 (2017), no. 3, 649–658. MR 3670886 |
| Reference:
|
[12] Yoon D. W., Lee C. W., Karacan M. K.: Some translation surfaces in the $3$-dimensional Heisenberg group.Bull. Korean Math. Soc. 50 (2013), no. 4, 1329–1343. MR 3092394, 10.4134/BKMS.2013.50.4.1329 |
| . |
