Önder Gökmen Yıldız, Mahmut ERGÜT, Mahmut AKYİĞİT

Helicoidal Surfaces Which Have the Timelike Axis in Minkowski Space with Density

In this paper, we study the prescribed curvature problem in manifold with density. We consider the Minkowski 3-space with a positive density function. For a given plane curve and an axis in the plane in Minkowski 3-space, a helicoidal surface can be constructed by the plane curve under helicoidal motions around the axis. Also we give examples of helicoidal surface with weighted Gaussian curvature.

Keywords:

Minkowski space Helicoidal, Manifold with density, Weighted curvature,

PDF

___

[1] C. Baba-Hamed, M. Bekkar, Helicoidal surfaces in the three-dimensional Lorentz-Minkowski space satisfying $\triangle^{II}r_{i}=\lambda_{i}r_{i}$}, J. Geom., 100(1-2) (2011), 1.
[2] M. P. Do Carmo, M. Dajczer, Helicoidal surfaces with constant mean curvature, Tohoku Math. J., Second Series, 34(3) (1982), 425-435.
[3] L. Rafael, E. Demir, Helicoidal surfaces in Minkowski space with constant mean curvature and constant Gauss curvature, Open Math., 12(9) (2014), 1349-1361.
[4] I. M. Roussos, The helicoidal surfaces as Bonnet surfaces. Tohoku Math. J., Second Series, 40(3)(1988), 485-490.
[5] C. Baikoussis, T. Koufogiorgos, Helicoidal surfaces with prescribed mean or Gaussian curvature, J. Geom., 63(1) (1998), 25-29.
[6] C. C. Beneki, G. Kaimakamis, B.J. Papantoniou, Helicoidal surfaces in three-dimensional Minkowski space, J. Math. Anal. Appl., 275(2) (2002), 586-614.
[7] F. Ji, Z. H. Hou, A kind of helicoidal surfaces in 3-dimensional Minkowski space, J. Math. Anal. Appl., 304(2) (2005), 632-643.
[8] D. W. Yoon, D. S. Kim, Y.H. Kim, J.W. Lee, Constructions of helicoidal surfaces in Euclidean space with Density, Symmetry, 9(9) (2017), 173.
[9] Ö. G. Yıldız, S. Hızal, M. Akyiğit, Type $I^{+}$ Helicoidal surfaces with prescribed weighted mean or Gaussian curvature in Minkowski space with density, An. Stiint. Univ. Ovidius Constant a, Seria Mat., 26(3) (2018), 99-108.
[10] F. Morgan, Manifolds with Density and Perelman’s Proof of the Poincaré Conjecture, Amer. Math. Monthly, 116(2) (2009), 134-142.
[11] D. T. Hıeu, N.M. Hoang, Ruled minimal surfaces in $\mathbb{R}^{3}$ with density $e^{z}$, Pacific J. Math., 243(2) (2009), 277-285.
[12] F. Morgan, Geometric measure theory: a beginner’s guide, Academic press, 2016.
[13] F. Morgan, Manifolds with density, Notices of the AMS, (2005), 853-858.
[14] F. Morgan, Myers’ theorem with density, Kodai Math. J., 29(3) (2006), 455-461.
[15] P. Rayon, M. Gromov, Isoperimetry of waists and concentration of maps, Geom. Funct. Anal., 13(1) (2003), 178-215.
[16] C. Rosales, A. Cañete, V. Bayle, M. Morgan, On the isoperimetric problem in Euclidean space with density, Calc. Var. Partial Differ. Equ., 31(1) (2008), 27-46.
[17] I. Corwin, N. Hoffman, S. Hurder, V. Šešum, Y. Xu, Differential geometry of manifolds with density, Rose-Hulman Undergrad. Math. J., 7(2006), 1-15.