Constant weighted mean curvature hypersurfaces in shrinking Ricci solitons

Abstract

In the mean curvature flow theory, a topic of great interest is to study possible singularitiesof this flow. In R n+1, the singularity models for this flow can be associated with hypersurfaces called f-minimal, that is, hypersurfaces with null weighted mean curvature. Some examples of f-minimal hypersurfaces are self-shrinkers, self-expanders and translating solitons, they play an important role in this theory since they describe singularity models for the mean curvature flow. In this thesis, we study a generalization of f-minimal hypersurfaces which are called CWMC hypersurfaces or λ-hypersurfaces in shrinking Ricci solitons. We prove some rigidity theorems seeking to classify these hypersurfaces in the Gaussian shrinking Ricci soliton and in the cylinder shrinking Ricci solitons. For the case the ambient is a cylinder shrinking Ricci soliton, we also study level sets and show some geometric properties of CWMC hypersurfaces.