ON WEAK BIHARMONIC GENERALIZED ROTATIONAL SURFACE IN E⁴

Abstract

In the present paper we consider weak biharmonic rotational surfaces in Euclidean 4-space E⁴. We have proved that the general rotational surface of parallel mean curvature vector field is weak biharmonic then either it is minimal or a constant mean curvature. Further, we show that if Vranceanu surface of constant mean curvature is weak-biharmonic then it is a Clifford torus in E⁴.

Keywords

• Biharmonic,Rotational surface

References

1. Referans1 K. Arslan, B. Kılıç Bayram, B. Bulca and G. Öztürk, Generalized Rotation Surfaces in E⁴. Results in Math. 61, 315--327 (2012).Referans2 : K. Arslan, R. Ezentas, C. Murathan and T. Sasahara, Biharmonic anti-invariant submanifolds in Sasakian space forms. Beitrage Algebra Geom. 48, 191--207 (2007). Referans3 : A. Balmus, S. Montaldo and C. Oniciuc, Classification results for biharmonic submanifolds in spheres. Israel J. Math. 168, 201--220 (2008). Referans4: A. Balmus, S. Montaldo and C. Oniciuc, Biharmonic Hypersurfaces in 4-Dimensional Space Forms. Math. Nachr. 283, 1696-1705 (2010). Referans5 : M. Barros and O.J. Garay, On submanifolds with harmonic mean curvature, Proc. Amer. Math. Soc. 129, 2545-2549 (1995). Referans6 : R. Caddeo, S. Montaldo and C. Oniciuc. Biharmonic submanifolds in spheres. Israel J. Math. 130, 109--123 (2002). Referans7 : B. Y. Chen,Geometry of Submanifolds, Dekker, New York (1973).Referans8: B.Y. Chen, A report on submanifolds of finite type. Soochow J. Math. 22, 117--337 (1996). Referans9 : B-Y. Chen and S. Ishikawa, Biharmonic surfaces in pseudo-Euclidean spaces, Memoirs of Fac. of Science, Kyushu University, Series A 45, 323-347 (1991). Referans10 : F. N. Cole, On rotations in space of four dimensions, Amer. J. Math. 12, 191-210 (1890). Referans11 : D.V. Cuong, Surfaces of Revolution with Constant Gaussian Curvature in Four-Space, Asian-Europian J. Math. 6 (2013). Referans12 : De Smet, D.J., Dillen F., Verstrealen L.and Vrancken L. A pointwise inequality in submanifold theory. Arc. Mat. (Bruno), 115-128 (1999).Referans13: F. Defever, Hypersurfaces of E⁴ with harmonic mean curvature vector field, Math. Nachr. 196, 61-69 1998). Referans14 : F. Defever, Bijdrageln tot de theorie van conform platte, semisymmetrische, en biharmonische deelvari ̈eteiten, Doctoral Thesis, Leuven (1999). Referans15 : I. Dimitric. Submanifolds of E^{m} with harmonic mean curvature vector, Bull. Inst.Math. Acad. Sinica, 20, 53-65 (1992). Referans16 : U. Dursun and N. C. Turgay, General rotational surfaces in Euclidean space E⁴ with pointwise 1-type Gauss map, Math. Com., 17, 71-81 (2012). Referans17 : G. Ganchev and V. Milousheva, On the Theory of Surfaces in the Four-dimensional Euclidean Space. Kodai Math. J., 31, 183-198 (2008).Referans18: Th. Hasanis and Th. Vlachos, Hypersurfaces in E⁴ with harmonic mean curvature vector field, Math. Nachr. 172, 145-169 (1995). Referans19 : B. Kiliç, K. Arslan , Ü. Lumiste and C. Murathan, On weak biharmonic submanifolds and 2-parallelity. Diff. Geo. Dyn. Sys. 5, 39-48 (2003). Referans20: D. Fetcu, E. Loubeau, S. Montaldo and C. Oniciuc, Biharmonic Submanifolds of Cⁿ. arXiv:0902.0268v1 [math.DG] 2 Feb 2009. Referans21: N. H. Kuiper, Minimal Total Absolute Curvature for Immersions. Invent. Math., 10, 209-238 (1970). Referans22: C. Moore, Surfaces of Rotations in a Space of Four Dimensions, Ann. Math. 2nd Ser., 21, 81-93 (1919). Referans23: Y.-L. Ou, Biharmonic hypersurfaces in Riemannian manifolds. arXiv:math.DG/09011507v1. Referans24: Y.-L. Ou, Some recent progress of Biharmonic Submanifolds, arXiv:1511.09103v1 [math.DG] 29 Nov 2015. Referans25: G. Vranceanu, Surfaces de Rotation dans E⁴, rev. Roum. Math. Pures Appl. XXII(6), 857-862 (1977). Referans26: Y.C. Wong, Contributions to the theory of surfaces in 4-space of constant curvature, Trans. Amer. Math. Soc, 59, 467-507 (1946). Referans27: D.W. Yoon, Some Properties of the Clifford Torus as Rotation Surfaces, Indian J. Pure Appl. Math. 34, 907-915 (2003).Referans28: S. T. Yau, Submanifolds with constant mean curvature, Amer. J. Math. 96,346--366 (1974).