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The Journal of Geometric Analysis

, Volume 29, Issue 1, pp 413–427 | Cite as

Complete Minimal Submanifolds with Nullity in the Hyperbolic Space

  • Marcos Dajczer
  • Theodoros Kasioumis
  • Andreas Savas-HalilajEmail author
  • Theodoros Vlachos
Article
  • 88 Downloads

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 \(\mathbb {H}^n\).

Keywords

Minimal submanifolds Index of relative nullity Real-analytic set Omori–Yau maximum principle 

Mathematics Subject Classification

53C42 53C40 

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  • Marcos Dajczer
    • 1
  • Theodoros Kasioumis
    • 2
  • Andreas Savas-Halilaj
    • 2
    Email author
  • Theodoros Vlachos
    • 2
  1. 1.IMPARio de JaneiroBrazil
  2. 2.Department of MathematicsUniversity of IoanninaIoanninaGreece

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