Maria Luiza Leite

Rotational hypersurfaces of space forms with constant scalar curvature

LetM be a complete rotational hypersurface of a space form with constant scalar curvatureS. In this paper we classify these hypersurfaces in the cases ofRn andHn, determine the admissible values ofS in each of the three spaces and give a geometrical description of the hypersurfaces according to the values ofS. In the case ofSn we find examples of embedded hypersurfaces with constantS∈(n−2/n−1, 1), which are not isometric to product of spheres.

Uniqueness and Nonexistence Theorems for Hypersurfaces with Hr = 0

We use the Maximum Principle to derive theorems of uniqueness and nonexistence for embedded, multiply connected and compact hypersurfaces with boundaries in parallel hyperplanes, which satisfy the geometric property Hr = 0.

How many maximal surfaces do correspond to one minimal surface

We discuss the minimal-to-maximal correspondence between surfaces and show that, under this correspondence, a congruence class of minimal surfaces in 3 determines an 2 -family of congruence classes of maximal surfaces in 3 . It is proved that further identifications among these classes may exist, depending upon the subgroup of automorphisms preserving the Hopf differential on the underlying Riemann surface. The space of maximal congruence classes inherits a quotient topology from 2 . In the case of the Scherk minimal surface, the subgroup has order two and the quotient space is topologically a disc with boundary. Other classical examples are discussed: for the Enneper minimal surface, one obtains a non-Hausdorff space; for the minimal catenoid, a closed interval.