Saddle surfaces in singular spaces

Kalikakis, Dimitrios

doi:10.1090/S0002-9947-04-03626-8

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.

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