Saddle surfaces in singular spaces
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- by Dimitrios E. Kalikakis PDF
- Trans. Amer. Math. Soc. 357 (2005), 2829-2841 Request permission
Abstract:
The notion of a saddle surface is well known in Euclidean space. In this work we extend the idea of a saddle surface to geodesically connected metric spaces. We prove that any solution of the Dirichlet problem for the Sobolev energy in a nonpositively curved space is a saddle surface. Further, we show that the space of saddle surfaces in a nonpositively curved space is a complete space in the Fréchet distance. We also prove a compactness theorem for saddle surfaces in spaces of curvature bounded from above; in spaces of constant curvature we obtain a stronger result based on an isoperimetric inequality for a saddle surface. These results generalize difficult theorems of S.Z. Shefel′ on compactness of saddle surfaces in a Euclidean space.References
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Additional Information
- Dimitrios E. Kalikakis
- Affiliation: Department of Mathematics, University of Crete, Irakleion, 714-09, Greece
- Email: kalikak@math.uoc.gr
- Received by editor(s): August 10, 2003
- Received by editor(s) in revised form: December 2, 2003
- Published electronically: October 28, 2004
- © Copyright 2004 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 357 (2005), 2829-2841
- MSC (2000): Primary 53C45, 53C43, 51M05
- DOI: https://doi.org/10.1090/S0002-9947-04-03626-8
- MathSciNet review: 2139929