Abstract
We deal with entire conformal Killing graphs, that is, graphs constructed through the flow generated by a complete conformal Killing vector field V on a Riemannian space \({\overline{M}}\), and which are defined over an integral leaf of the foliation \({V^\bot {\rm of} \overline{M}}\) orthogonal to V. Under a suitable restriction on the norm of the gradient of the function z which determines such a graph Σ(z), we establish sufficient conditions to ensure that Σ(z) is totally geodesic. Afterwards, when the ambient space \({\overline{M}}\) has constant sectional curvature, we obtain lower estimates for the index of minimum relative nullity of Σ(z).
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References
Alexandrov A.D.: Uniqueness theorems for surfaces in the large I. Vestinik Leiningrad Univ. 11, 5–17 (1956)
Alexandrov A.D.: A characteristic property of spheres. Ann. Mat. Pura Appl. 58, 303–315 (1962)
Alías L.J., Dajczer M., Ripoll J.B.: A Bernstein-type theorem for Riemannian manifolds with a Killing field. Ann. Glob. Anal. Geom. 31, 363–373 (2007)
Alías L.J., Impera D., Rigoli M.: Hypersurfaces of constant higher order mean curvature in warped products. Trans. Am. Math. Soc. 365, 591–621 (2013)
Alías L.J., de Lira J.H., Malacarne J.M.: Constant higher order mean curvature hypersufaces in Riemannian spaces. J. Inst. Math. Jussieu 5, 527–562 (2006)
Andrzejewski K., Walczak P.: The Newton transformation and new integral formulae for foliated manifolds. Ann. Global Anal. Geom. 37, 103–111 (2010)
Barros A., Sousa P.: Compact graphs over a sphere of constant second order mean curvature. Proc. Am. Math. Soc. 137, 3105–3114 (2009)
Bernstein S.: Sur les surfaces d’efinies au moyen de leur courboure moyenne ou totale. Ann. Ec. Norm. Sup. 27, 233–256 (1910)
Caminha A.: On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds. J. Geom. Phys. 56, 1144–1174 (2006)
Caminha A.: The geometry of closed conformal vector fields on Riemannian spaces. Bull. Brazilian Math. Soc. 42, 277–300 (2011)
Caminha A., de Lima H.F.: Complete vertical graph with constant mean curvature in Semi-Riemannian warped products. Bull. Belgian Math. Soc. 16, 91–105 (2009)
Dajczer M. et al.: Submanifolds and Isometric Immersions. Publish or Perish, Houston (1990)
Dajczer M., Hinojosa P., de Lira J.H.: Killing graphs with prescribed mean curvature. Calc. Var. Partial Diff. Eq. 33, 231–248 (2008)
Dajczer M., de Lira J.H.: Conformal Killing graphs with prescribed mean curvature. J. Geom. Anal. 22, 780–799 (2012)
Ferus D.: On the completeness of nullity foliations. Michigan Math. J. 18, 61–64 (1971)
Jellett J.J.: Sur la surface dont la courbure moyenne est constante. J. Math. Pures Appl. 18, 163–167 (1853)
Liebmann, H.: Eine neue Eigenschaft der Kugel. Nachr. Kg. Ges. Wiss. Götingen, Math. Phys. Kl., 44–55 (1899)
Montiel S.: Unicity of constant mean curvature hypersurfaces in some Riemannian manifolds. Indiana Univ. Math. J. 48, 711–748 (1999)
Pan T.K.: Conformal vector fields in compact Riemannian manifolds. Proc. Am. Math. Soc. 14, 653–657 (1963)
Yau S.T.: Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Univ. Math. J. 25, 659–670 (1976)
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de Lima, H.F., de Lima, J.R. & Velásquez, M.A.L. On the nullity of conformal Killing graphs in foliated Riemannian spaces. Aequat. Math. 87, 285–299 (2014). https://doi.org/10.1007/s00010-013-0203-0
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DOI: https://doi.org/10.1007/s00010-013-0203-0