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Annals of Global Analysis and Geometry

, Volume 56, Issue 1, pp 137–146 | Cite as

Vanishing theorems for the cohomology groups of free boundary submanifolds

  • Marcos P. CavalcanteEmail author
  • Abraão Mendes
  • Feliciano Vitório
Article
  • 62 Downloads

Abstract

In this paper, we prove that there exists a universal constant C, depending only on positive integers \(n\ge 3\) and \(p\le n-1\), such that if \(M^n\) is a compact free boundary submanifold of dimension n immersed in the Euclidean unit ball \(\mathbb {B}^{n+k}\) whose size of the traceless second fundamental form is less than C, then the pth cohomology group of \(M^n\) vanishes. Also, employing a different technique, we obtain a rigidity result for compact free boundary surfaces minimally immersed in the unit ball \(\mathbb {B}^{2+k}\).

Keywords

Freeboundary submanifolds Second fundamental form Harmonic forms Cohomology groups 

Mathematics Subject Classification

Primary 53C40 58A10 

Notes

Acknowledgements

The authors are grateful to Levi Lima and Ezequiel Barbosa for their interest and helpful discussions about this work. The authors were partially supported by CNPq-Brazil, CAPES-Brazil and FAPEAL-Brazil.

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal de AlagoasMaceióBrazil

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