Topological and rigidity results for four-dimensional hypersurfaces in space forms

Exploiting the special features of four-dimensional Riemannian geometry, we derive topological and rigidity results for hypersurfaces immersed in space forms of dimension 5. First, we provide a complete description of the Weyl tensor for four-dimensional hypersurfaces, by means of which we derive a new characterization result for isoparametric hypersurfaces; then, we prove sharp topological bounds on the Weyl functional for closed, minimal hypersurfaces, involving the Euler characteristic in the case of an ambient space with constant non-negative sectional curvature. Then, inspired by a famous conjecture by Chern and the so-called second pinching problem, we find estimates for the norm of the second fundamental form in terms of the Euler characteristic in the minimal, constant scalar curvature case, under a cross-sectional area assumption. Finally, we prove some rigidity results by means of integral inequalities on the derivatives of the second fundamental form, also dealing with special curvature conditions, such as half harmonic Weyl curvature and Bach-flatness. We also extend some of the local results to the case of a locally conformally flat 5-dimensional ambient space.