ON THE GEOMETRY OF NEWTON OPERATORS

We studied the geometry of hypersurfaces of complete constant higher order mean curvature, both in the Riemannian and in the Lorentzian setting. In particular, in the Riemannian setting, we focused on uniqueness results for hypersurfaces in warped products. An analytic approach based on a general version of the Omori-Yau maximum principle for trace-type semi-elliptic operators and on the parabolicity for elliptic operators in divergence form, combined with suitable geometric conditions on the geometry of the ambient manifold, allowed to characterize the slices as the only hypersurfaces of constant higher order mean curvature in warped products. Later on, we studied how to extend, using the same technique, uniqueness results to spacelike hypersurfaces of constant higher order mean curvature in Lorentzian warped products. Finally, we focused on comparison geometry in the Lorentzian setting proving, under suitable bounds on the Ricci or the sectional curvature of a Lorentzian manifold, hessian and laplacian comparison theorems for the Lorentzian distance function. Jointly with the Omori-Yau maximum principle, these theorems, applied to the distance function restricted to spacelike hypersurfaces, allowed to obtain higher order mean curvature estimates for spacelike hypersurfaces bounded by a level set of the distance function and Bernstein-type theorems.