Geometric properties of self-shrinkers in cylinder shrinking Ricci   solitons
  In this paper we prove some spectral properties of the drifted Laplacian of self-shrinkers properly immersed in gradient shrinking Ricci solitons. Then we use these results to prove some geometric properties of self-shrinkers. For example, we describe a collection of domains in the ambient space that cannot contain self-shrinkers. 
Matheus Vieira, Detang Zhou
math.DG