Saddle Surfaces

The class of saddle surfaces is dual to the class of convex surfaces. A surface in a Euclidean space is saddle if it is impossible to cut off a crust by any hyperplane. Historically, saddle surfaces are connected with the Plateau problem. In this book we extend the idea of a saddle surface to geodesically connected metric spaces and we study the class of saddle surfaces in spaces of curvature bounded from above in the sense of A.D. Aleksandrov. Metric spaces of bounded curvature inherit basic properties of Riemannian spaces while they may have strong topological and metric singularities. For convenience we first review the basic theory of metric spaces of curvature bounded from above in the sense of Aleksandrov. Then we provide a source of examples and we classify several properties of saddle surfaces including completeness and compactness theorems, optimization problems, isoperimetric inequalities and issues about their intrinsic curvature. Stronger results are presented in spaces of constant curvature.