Advertisement

The Morse index of the critical catenoid

  • Graham Smith
  • Detang Zhou
Original Paper
  • 7 Downloads

Abstract

We show that the rotationally symmetric free boundary minimal catenoid in the unit ball in \({\mathbb {R}}^3\) has Morse index equal to 4.

Keywords

Free boundary minimal surfaces Morse index 

Mathematics Subject Classification

53A10 

References

  1. 1.
    Brendle, S.: Embedded minimal tori in \(S^3\) and the Lawson conjecture. Acta Math. 211(2), 177–190 (2013)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Chen, J., Fraser, A., Pang, C.: Minimal immersions of compact bordered Riemann surfaces with free boundary. Trans. Am. Math. Soc. 367, 2487–2507 (2015)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Codá, Marques F., Neves, A.: Min–max theory and the Willmore conjecture. Ann. Math. 179(2), 683–782 (2014)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Devyver, B.: Index of the critical catenoid. arXiv:1609.02315
  5. 5.
    Fraser, A., Schoen, R.: The first Steklov eigenvalue, conformal geometry, and minimal surfaces. Adv. Math. 226(5), 4011–4030 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fraser, A., Schoen, R.: Sharp eigenvalue bounds and minimal surfaces in the ball. Invent. Math. 203(3), 823–890 (2016)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Máximo, D., Nunes, I., Smith, G.: Free boundary minimal annuli in convex three-manifolds. J. Diff. Geom. 106(1), 139–186 (2017)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Tranh, H.T.: Index characterization for free boundary minimal surfaces. arXiv:1609.01651
  9. 9.
    Urbano, F.: Minimal surfaces with low index in the three-dimensional sphere. Proc. Am. Math. Soc. 108, 989–992 (1990)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Instituto de Matemática, UFRJRio de JaneiroBrazil
  2. 2.Instituto de Matemática, UFFNiteróiBrazil

Personalised recommendations