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Saddle surfaces in singular spaces
Author(s):
Dimitrios
E.
Kalikakis
Journal:
Trans. Amer. Math. Soc.
357
(2005),
2829-2841.
MSC (2000):
Primary 53C45, 53C43, 51M05
Posted:
October 28, 2004
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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:
10.1090/S0002-9947-04-03626-8
PII:
S 0002-9947(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
Posted:
October 28, 2004
Copyright of article:
Copyright
2004,
American Mathematical Society
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