Olin I. Some comparison theorems for convex surfaces in simply connected Finsler spaces of non-positive flag curvature

The thesis is devoted to the study of convex surfaces in simply connected Finsler spaces of non-positive flag curvature. Locally convex compact immersed hypersurfaces in Finsler-Hadamard manifolds with bounded T-curvature are considered. It is proved that every such hypersurface is embedded as the boundary of some convex body under certain conditions on the normal curvatures. The upper and lower estimates for the ratio between volume of metric ball and area of its surface in Finsler-Hadamard manifolds and Hilbert geometry are given. It is shown that the spheres in Hilbert geometry have the same volume growth entropy as those in the hyperbolic space. The tendency to 1 of the normal curvatures of the spheres centered at the same point and Rund and Finsler curvatures of the circle in Hilbert geometry as the radii tend to infinity are proved.