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Annals of Global Analysis and Geometry

, Volume 54, Issue 2, pp 273–299 | Cite as

Morse theory for minimal surfaces in manifolds

  • Hwajeong Kim
Article
  • 91 Downloads

Abstract

A Morse theory of a given function gives information of the numbers of critical points of some topological type. A minimal surface, bounded by a given curve in a manifold, is characterized as a harmonic extension of a critical point of the functional \({\mathcal E}\) which corresponds to the Dirichlet integral. We want to obtain Morse theories for minimal surfaces in Riemannian manifolds. We first investigate the higher differentiabilities of \({\mathcal E}\). We then develop a Morse inequality for minimal surfaces of annulus type in a Riemannian manifold. Furthermore, we also construct body handle theories for minimal surfaces of annulus type as well as of disc type. Here we give a setting where the functional \({\mathcal E}\) is non-degenerated.

Keywords

Minimal surfaces Harmonic maps Abstract critical point theory Morse theory 

Mathematics Subject Classification

49Q05 58E20 58E05 37B30 

Notes

Acknowledgements

This research was supported by Hannam University in 2016.

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Copyright information

© Springer Science+Business Media B.V., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsHannam UniversityDaejeonRepublic of Korea

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