Método conforme aplicado ao problema de Bernstein estável e problemas de Pinching

Abstract

The goal of this work is to study applications of the conformal method to the stable Bernstein problem and pinching problems for immersions in the sphere. First, we address the results of Catino, Mastrolia & Roncoroni [4], who, through an appropriate conformal change, prove u solution for the stable Bernstein problem in dimension n = 3: the complete, orientable, stable, minimal hypersurfaces immersed in R4 are hyperplanes. Furthermore, using the ideas from the same work, we will show that there are no stable minimal hypersurfaces immersed in complete Riemannian manifolds of dimension n  6 with non-negative sectional curvature and uniformly positive Ricci curvature. Next, we discuss the results of Magliaro, Mari, Roing & Savas-Halilaj [26], who, also relying on an appropriate conformal change, prove some pinching results for complete submanifolds immersed in the sphere. In particular, this technique generalizes classical pinching results that are verified for compact submanifolds, such as those in [7], [1], and [30].