Abstract
We investigate complete minimal submanifolds \(f: M^3\rightarrow \mathbb {H}^n\) in hyperbolic space with index of relative nullity at least one at any point. The case when the ambient space is either the Euclidean space or the round sphere was already studied in Dajczer et al. (Math Z 287: 481–491, 2017 and Comment Math Helv, to appear, 2017), respectively. If the scalar curvature is bounded from below we conclude that the submanifold has to be either totally geodesic or a generalized cone over a complete minimal surface lying in an equidistant submanifold of \(\mathbb {H}^n\).
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References
Alias, L., Mastrolia, P., Rigoli, M.: Maximum Principles and Geometric Applications. Springer Monographs in Mathematics. Springer, Cham (2016)
Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc 145, 1–49 (1969)
Carberry, E., Turner, K.: Harmonic tori in De Sitter spaces \(\mathbb{S}_1^{2n}\). Geom. Dedic. 170, 143–155 (2014)
Dajczer, M., Gromoll, D.: Rigidity of complete Euclidean hypersurfaces. J. Differ. Geom. 31, 401–416 (1990)
Dajczer, M., et al.: Submanifolds and Isometric Immersions. Math. Lecture Ser., vol. 13. Publish or Perish Inc., Houston (1990)
Dajczer, M., Kasioumis, Th, Savas-Halilaj, A., Vlachos, Th: Complete minimal submanifolds with nullity in Euclidean space. Math. Z. 287, 481–491 (2017)
Dajczer, M., Kasioumis, Th., Savas-Halilaj, A., Vlachos, Th.: Complete minimal submanifolds with nullity in Euclidean spheres. Comment. Math. Helv., to appear (2018)
Eells, J., Sampson, J.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)
Ejiri, N.: Isotropic harmonic maps of Riemannian surfaces into De Sitter space time. Q. J. Math. 39, 291–306 (1988)
Hasanis, Th, Savas-Halilaj, A., Vlachos, Th: Complete minimal hypersurfaces in the hyperbolic space \(\mathbb{H}^4\) with vanishing Gauss-Kronecker curvature. Trans. Am. Math. Soc. 359, 2799–2818 (2007)
Hulett, E.: Superconformal harmonic surfaces in De Sitter space–times. J. Geom. Phys. 55, 179–206 (2005)
Kasioumis, Th.: Ruled minimal submanifolds with rank at most four in the hyperbolic space. In preparation (2018)
Krantz, S., Parks, H.: A Primer of Real Analytic Functions. Birkhäuser Advanced Texts. Basler Lehrbücher, Birkhäuser Boston Inc., Boston (2002)
Meier, M.: Removable singularities of harmonic maps and an application to minimal submanifolds. Indiana Univ. Math. J. 35, 705–726 (1986)
Taylor, M.: Partial Differential Equations I. Basic theory. Applied Mathematical Sciences, vol. 115, 2nd edn. Springer, New York (2011)
Yau, S.-T.: Harmonic functions on complete Riemannian manifolds. Commun. Pure Appl. Math. 28, 201–228 (1975)
Wood, C.M.: The energy of a unit vector field. Geom. Dedic. 64, 319–330 (1997)
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This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. This work was partially supported by MINECO/FEDER Project Reference MTM2015-65430-P and Fundación Séneca Project Reference 19901/GERM/15, Spain.
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Dajczer, M., Kasioumis, T., Savas-Halilaj, A. et al. Complete Minimal Submanifolds with Nullity in the Hyperbolic Space. J Geom Anal 29, 413–427 (2019). https://doi.org/10.1007/s12220-018-9998-1
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DOI: https://doi.org/10.1007/s12220-018-9998-1