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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Saddle surfaces in singular spaces


Author: Dimitrios E. Kalikakis
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
Published electronically: October 28, 2004
MathSciNet review: 2139929
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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.


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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

DOI: https://doi.org/10.1090/S0002-9947-04-03626-8
Keywords: Saddle surface, CAT($\kappa$) space, curvature in the sense of A.D. Aleksandrov, isoperimetric inequality, compactness
Received by editor(s): August 10, 2003
Received by editor(s) in revised form: December 2, 2003
Published electronically: October 28, 2004
Article copyright: © Copyright 2004 American Mathematical Society