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, Volume 158, Issue 1–2, pp 21–30 | Cite as

Rigidity of complete minimal submanifolds in a hyperbolic space

  • Hudson Pina de Oliveira
  • Changyu XiaEmail author


In this paper we prove some gap theorem for complete immersed minimal submanifold of dimension no less than six or four, depending on the codimension, in a hyperbolic space \(\mathbb {H}^{n+m}(-1)\). That is, we show that a high dimensional complete immersed minimal submanifold M in \( \mathbb {H}^{n+m}(-1)\), is totally geodesic if the \(L^d\) norm of |A|, for some d, on geodesic balls centered at some point \(p \in M \) has less than quadratic growth and if either \(\sup _{x \in M} |A|^2\) is not too large or the \(L^n\) norm of |A| on M is finite, were, A is the second fundamental form of M.

Mathematics Subject Classification

53C21 53C25 


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© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Universidade Federal de Mato Grossocampus AraguaiaBarra do Garças-MTBrazil
  2. 2.Universidade de BrasíliaBrasíliaBrazil

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