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Keywords:
Laplacian operator; homogeneous space; invariant surface; surfaces of coordinate finite type
Laplacian operator; homogeneous space; invariant surface; surfaces of coordinate finite type
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.
References:
[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
[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. DOI 10.1007/s00022-012-0136-0 | MR 3017050
[3] Chen B.-Y.: Total Mean Curvature and Submanifolds of Finite Type. Series in Pure Mathematics, 1, World Scientific Publishing, Singapore, 1984. MR 0749575 | Zbl 0537.53049
[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.
[5] Inoguchi J., López R., Munteanu M.-I.: Minimal translation surfaces in the Heisenberg group $ Nil_{3}$. Geom. Dedicata 161 (2012), 221–231. DOI 10.1007/s10711-012-9702-8 | MR 2994039
[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. DOI 10.2206/kyushujm.65.237 | MR 2977760
[7] López R., Munteanu M. I.: Minimal translation surfaces in $ Sol_{3}$. J. Math. Soc. Japan. 64 (2012), no. 3, 985–1003. DOI 10.2969/jmsj/06430985 | MR 2965436
[8] Scott P.: The geometries of $3$-manifolds. Bull. London Math. Soc. 15 (1983), no. 5, 401–487. DOI 10.1112/blms/15.5.401 | MR 0705527 | Zbl 0662.57001
[9] Takahashi T.: Minimal immersions of Riemannian manifolds. J. Math. Soc. Japan 18 (1966), 380–385. DOI 10.2969/jmsj/01840380 | MR 0198393
[10] Troyanov M.: L'horizon de $ SOL$. Exposition. Math. 16 (1998), no. 5, 441–479. MR 1656902
[11] Yoon D. W.: Coordinate finite type invariant surfaces in $ Sol$ spaces. Bull. Iranian Math. Soc. 43 (2017), no. 3, 649–658. MR 3670886
[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. DOI 10.4134/BKMS.2013.50.4.1329 | MR 3092394
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