Conformally flat Willmore spacelike hypersurfaces in Rn+11
In this paper, we give the equation satisfied by umbilics-free Willmore spacelike hypersurfaces using the conformal invariants in Lorentzian space forms. At the same time, we give the equation satisfied by hyperelastic spacelike curves in 2-dimensional Lorentzian space forms and classify the closed hyperelastic spacelike curves. Finally conformally flat Willmore spacelike hypersurfaces are classified in terms of the hyperelastic spacelike curves in 2-dimensional Lorentzian space forms.
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- [1] Barros M, Ferrández A, Lucas P, Meroño M. Willmore tori and Willmore-chen submanifolds in pseudo-Riemannian
Spaces. Journal of Geometry and Physics 1998; 28: 45-66.
- [2] Cahen M, Kerbrat Y. Domaines symmétriques des quadriques projectives. Journal de Mathématiques Pures et
Appliquées 1983; 62: 327-348 (in French).
- [3] Dussan MP, Magid M. Conformally flat Lorentz hypersurfaces in the conformal compactification of Lorentz space.
Journal of Geometry and Physics 2007; 57: 2466-2482.
- [4] Guo Z. Willmore submanifolds in the unite sphere. Collectanea Mathematica 2004; 5 (3): 279-287.
- [5] Guo Z, Li H. Conformal invariants of submanifolds in a Riemannian space and conformal rigidity theorems on
Willmore hypersurfaces. Journal of Geometric Analysis 2018; 28 (3): 2670-2691.
- [6] Guo Z, Li HZ, Wang CP. The second variational formula for Willmore submanifolds. Results in Mathematics 2001;
40: 205-225.
- [7] Guo Z, Li HZ, Wang CP. The Möbius characterizations of Willmore tori and Veronese submanifolds in unit sphere.
Pacific Journal of Mathematics 2009; 241: 227-242.
- [8] Kobayashi S, Nomizu K. Foundations os Differential Geometry (I). New York, NY, USA: Wiley Interscience, 1969.
- [9] Li HZ. Willmore hypersurfaces in a sphere. Asian Journal of Mathematics 2001; 5: 365-378.
- [10] Li HZ. Willmore surfaces in Sn. Annals of Global Analysis and Geometry 2002; 21: 203-213.
- [11] Li TZ. Willmore hypersurfaces with two distinct principal curvatures in R
n+1 . Pacific Journal of Mathematics
2012, 256: 129-149.
- [12] Li TZ, Ma X, Wang CP. Willmore hypersurfaces with constant Möbius curvature in R
n+1 . Geometriae Dedicata
2013; 166: 251-267.
- [13] Li TZ, Nie CX. Spacelike Dupin hypersurfaces in Lorentzian space forms. Journal of the Mathematical Society of
Japan 2018; 70: 463-480.
- [14] Marques FC, Neves A. Min-max theory and the Willmore conjecture. Annals of Mathematics 2014; 79: 683-782.
- [15] O’Neil B. Semi-Riemannian Geometry. New York, NY, USA: Academic Press, 1983.
- [16] Wang CP. Möbius geometry of submanifolds in S
n
. Manuscripta Mathematica 1998; 96: 517-534.