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


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 \(\mathbb {B}^{2+k}\).


Freeboundary submanifolds Second fundamental form Harmonic forms Cohomology groups 

Mathematics Subject Classification

Primary 53C40 58A10 



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.


  1. 1.
    Alencar, H., do Carmo, M.: Hypersurfaces with constant mean curvature in spheres. Proc. Am. Math. Soc. 120(4), 1223–1229 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Ambrozio, L., Carlotto, A., Sharp, B.: Index estimates for free boundary minimal hypersurfaces. Math. Ann. 370(3–4), 1063–1078 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Ambrozio, L., Nunes, I.: A gap theorem for free boundary minimal surfaces in the three-ball. arXiv:1608.05689 [math.DG]. To appear in Communications in Analysis and Geometry
  4. 4.
    An-Min, L., Jimin, L.: An intrinsic rigidity theorem for minimal submanifolds in a sphere. Arch. Math. (Basel) 58(6), 582–594 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Batista, M., Mirandola, H., Vitório, F.: Hardy and Rellich inequalities for submanifolds in Hadamard spaces. J. Differ. Equ. 263(9), 5813–5829 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Calderbank, D.M.J., Gauduchon, P., Herzlich, M.: Refined Kato inequalities and conformal weights in Riemannian geometry. J. Funct. Anal. 173(1), 214–255 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Chern, S.S., do Carmo, M., Kobayashi, S.: Minimal submanifolds of a sphere with second fundamental form of constant length, Functional Analysis and Related Fields (Proc. Conf. for M. Stone, Univ. Chicago, Chicago, Ill., 1968), Springer, New York, (1970), pp. 59–75Google Scholar
  8. 8.
    de Lima, L.L.: A Feynman–Kac formula for differential forms on manifolds with boundary and geometric applications. Pac. J. Math. 292(1), 177–201 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Flaherty, F.: Spherical submanifolds of pinched manifolds. Am. J. Math. 89, 1109–1114 (1967)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Fraser, A., Schoen, R.: Uniqueness theorems for free boundary minimal disks in space forms. Int. Math. Res. Not. IMRN 1(17), 8268–8274 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Herzlich, M.: Refined Kato inequalities in Riemannian geometry, Journées “Équations aux Dérivées Partielles” (La Chapelle sur Erdre, 2000), University of Nantes, Nantes, pp. Exp. No. VI, 11 (2000)Google Scholar
  12. 12.
    Howard, R., Wei, S.W.: Inequalities relating sectional curvatures of a submanifold to the size of its second fundamental form and applications to pinching theorems for submanifolds. Proc. Am. Math. Soc. 94(4), 699–702 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lawson Jr., H.B.: Local rigidity theorems for minimal hypersurfaces. Ann. Math. (2) 89, 187–197 (1969)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lawson Jr., H.B., Simons, J.: On stable currents and their application to global problems in real and complex geometry. Ann. Math. (2) 98, 427–450 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Leung, P.F.: Minimal submanifolds in a sphere. Math. Z. 183(1), 75–86 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Leung, P.F.: On a relation between the topology and the intrinsic and extrinsic geometries of a compact submanifold. Proc. Edinb. Math. Soc. (2) 28(3), 305–311 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lin, H.: On the structure of submanifolds in Euclidean space with flat normal bundle. Results Math. 68(3–4), 313–329 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Shiohama, K., Hongwei, X.: The topological sphere theorem for complete submanifolds. Compositio Math. 107(2), 221–232 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Simons, J.: Minimal varieties in riemannian manifolds. Ann. Math. (2) 88, 62–105 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Xu, H.W.: A rigidity theorem for submanifolds with parallel mean curvature in a sphere. Arch. Math. (Basel) 61(5), 489–496 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Yano, K.: Integral Formulas in Riemannian Geometry, Pure and Applied Mathematics, vol. 1. Marcel Dekker, Inc., New York (1970)Google Scholar
  22. 22.
    Yau, S.T.: Submanifolds with constant mean curvature. I, II, Am. J. Math. 96, 346–366 (1974)MathSciNetzbMATHGoogle Scholar
  23. 23.
    Yau, Shing Tung: Submanifolds with constant mean curvature. II Amer. J. Math. 97, 76–100 (1975)CrossRefzbMATHGoogle Scholar

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Instituto de MatemáticaUniversidade Federal de AlagoasMaceióBrazil

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