Drach K. Extreme bounds for complete hypersurfaces in Riemannian spaces

In the thesis we study some extreme properties of complete strictly convex hypersurfaces whose normal curvatures are uniformly bounded from below by a positive constant, or pinched with two non-negative constants, and their non-smooth generalizations. We prove the radial angle comparison theorem for angles between the normal vector field of a complete embedded strictly convex hypersurface in Riemannian manifolds of bounded from below sectional curvature and the radial vector field with respect to a given fixed point inside the domain bounded by the surface. We also obtain sharp bounds for these angles. Similar results are proved in Lorentzian manifolds. The connection between the radial angle comparison theorem and Blaschke’s Rolling Theorem is investigated. We find sharp bounds on the width and the quotient of the radii of a spherical shell in which one can put a complete embedded strictly convex hypersurface in Riemannian spaces. In the dissertation on the 2-planes of constant curvature for closed embedded strictly convex curves we completely solve the reverse isoperimetric problem of finding a curve that encloses the smallest area among the curves with a given length. The corresponding reverse isoperimetric inequalities are also obtained.