Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 40, 2015, 659-668

RIGIDITY OF COMPLETE MINIMAL HYPERSURFACES IN A HYPERBOLIC SPACE

Nilton M. Barroso Neto, Qiaoling Wang and Changyu Xia

Universidade de Brasilia, Departamento de Matemática
70910-900, Brasilia - DF, Brazil; n.m.b.neto 'at' mat.unb.br

Universidade de Brasilia, Departamento de Matemática
70910-900, Brasilia - DF, Brazil; wang 'at' mat.unb.br

Universidade de Brasilia, Departamento de Matemática
70910-900, Brasilia - DF, Brazil; xia 'at' mat.unb.br

Abstract. This paper provides a gap theorem for the first eigenvalue of the stability operator of complete immersed minimal hypersurfaces of dimension no less than five in a hyperbolic space. Namely, we show that an n(≥ 5)-dimensional complete immersed minimal hypersurface M in a hyperbolic space is totally geodesic if the first eigenvalue of the stability operator of M is bigger than some concrete constant and if the L² norm of the length of the second fundamental form of M grows properly.

2010 Mathematics Subject Classification: Primary 53C20, 53C42.

Key words: Minimal hypersurface, first eigenvalue, stability operator, hyperbolic space, rigidity.

Reference to this article: N.M.B. Neto, Q. Wang and C. Xia: Rigidity of complete minimal hypersurfaces in a hyperbolic space. Ann. Acad. Sci. Fenn. Math. 40 (2015), 659-668.

Full document as PDF file

doi:10.5186/aasfm.2015.4036