Mathematische Zeitschrift

, Volume 292, Issue 1–2, pp 513–527 | Cite as

Quantitative index bounds for translators via topology

  • Debora Impera
  • Michele RimoldiEmail author


We obtain a quantitative estimate on the generalised index of translators for the mean curvature flow with bounded norm of the second fundamental form. The estimate involves the dimension of the space of weighted square integrable f-harmonic 1-forms. By the adaptation to the weighted setting of Li–Tam theory developed in previous works, this yields estimates in terms of the number of ends of the hypersurface when this is contained in a upper halfspace with respect to the translating direction. When there exists a point where all principal curvatures are distinct we estimate the nullity of the stability operator. This permits to obtain quantitative estimates on the stability index via the topology of translators with bounded norm of the second fundamental form which are either two-dimensional or (in higher dimension) have finite topological type and are contained in a upper halfspace.


Translators Index estimates Genus Number of ends 

Mathematics Subject Classification

53C42 53C21 



The authors are deeply grateful to Alessandro Savo for his interest in this work and a number of enlightening discussions. The first author is partially supported by INdAM-GNSAGA. The second author acknowledge partial support by INdAM-GNAMPA.


  1. 1.
    Ambrozio, L., Carlotto, A., Sharp, B.: A note on the index of closed minimal hypersurfaces of flat tori. Proc. Am. Math. Soc. 146(1), 335–344 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Bueler, E.L.: The heat kernel weighted Hodge Laplacian on noncompact manifolds. Trans. Am. Math. Soc. 351(2), 683–713 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Carron, G.: Inégalités de Sobolev et \(L^2\)-cohomologie, Séminaire de Théorie Spectrale et Géométrie, Année, 13, 171–176 (1994–1995) (Univ. Grenoble I, Saint-Martin-d’Hères)Google Scholar
  4. 4.
    Carron, G.: \(L^{2}\) harmonic forms on non-compact manifolds. Lecture notes. Accessed 2007
  5. 5.
    Cheng, X., Zhou, D.: Stability properties and gap theorem for complete \(f\)-minimal hypersurfaces. Bull. Braz. Math. Soc. (N. S.) 46(2), 251–274 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Dávila, J., del Pino, M., Nguyen, X.H.: Finite topology self-translating surface for the mean curvature flow in \(\mathbb{R}^{3}\). Adv. Math. 320, 674–729 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Farkas, H.M., Kra, I.: Riemann surfaces, Graduate Texts in Mathematics, vol. 71, p. xvi+363. Springer, New York (1992)zbMATHGoogle Scholar
  8. 8.
    Gaffney, M.P.: The heat equation of Milgram and Rosenbloom for open Riemannian manifolds. Ann. Math. (2) 60, 458–466 (1954)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hoffman, D., Spruck, J.: Sobolev and isoperimetric inequalities for Riemannian submanifolds. Comm. Pure Appl. Math. 27, 715–727 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kunikawa, K., Saito, S.: Remarks on topology of stable translating solitons. arXiv preprint server, arXiv:1804.05436
  11. 11.
    Impera, D., Rimoldi, M.: Stability properties and topology at infinity of \(f\)-minimal hypersurfaces Geom. Dedicata 178, 21–47 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Impera, D., Rimoldi, M.: Rigidity results and topology at infinity of translating solitons of the mean curvature flow. Commun. Contemp. Math. 19(6), 1750002 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Impera, D., Rimoldi, M., Savo, A.: Index and first Betti number of \(f\)-minimal hypersurfaces and self-shrinkers (To appear on Rev. Mat. Iberoam). arXiv:1803.08268 (preprint, ArXiv Preprint Server)
  14. 14.
    Li, C.: Index and topology of minimal hypersurfaces of \(\mathbb{R}^{n}\). Calc. Var. Partial Differ. Equ. 56(6), 18 (2017)Google Scholar
  15. 15.
    Nguyen, X.H.: Translating tridents. Commun. Partial Differ. Equ. 34(1–3), 257–280 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Nguyen, X.H.: Complete embedded self-translating surfaces under mean curvature flow. J. Geom. Anal. 23(3), 1379–1426 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Palmer, B.: Stability of minimal hypersurfaces. Comment. Math. Helv. 66(2), 185–188 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ros, A.: One-sided complete stable minimal surfaces. J. Differ. Geom. 74(1), 69–92 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Savo, A.: Index bounds for minimal hypersurfaces of the sphere. Indiana Univ. Math. J. 59(3), 823–837 (2010)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Dipartimento di Scienze Matematiche “Giuseppe Luigi Lagrange”Politecnico di TorinoTurinItaly

Personalised recommendations