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 \(\mathbb {B}^{2+k}\).
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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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Cavalcante, M.P., Mendes, A. & Vitório, F. Vanishing theorems for the cohomology groups of free boundary submanifolds. Ann Glob Anal Geom 56, 137–146 (2019). https://doi.org/10.1007/s10455-019-09660-1
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DOI: https://doi.org/10.1007/s10455-019-09660-1