Geometric Properties of Self-Shrinkers in Cylinder Shrinking Ricci Solitons

Abstract

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.

Parabolicity criteria and characterization results for submanifolds of bounded mean curvature in model manifolds with weights

“…From the calculus in Example 4.1 the critical ball for n-dimensional self-shrinkers in R m is B √ n . We remark that properly immersed self-shrinker hypersurfaces inside some Euclidean balls were described by Vieira and Zhou [59,Thm. 1].…”

Uniqueness results and enclosure properties for hypersurfaces with boundary in weighted cylinders

“…As a consequence, we deduce half-space and Bernstein type theorems for hypersurfaces with non-empty boundary in some weighted manifolds. Related results for hypersurfaces with empty boundary have been established by many authors in several previous works, see for instance [41,32,20,38,10,9,30,8,7,21,16,14,40,13,1,15,2,19,27,33]. We also generalize to the weighted context some well-known enclosure properties for compact minimal surfaces with boundary in R 3 , like the convex hull property, the hyperboloid theorem and the cone theorem, see [17,Ch.…”

Minimal submanifolds in a metric measure space

“…Also in [5], Meija and we obtained the equivalence of properness of immersion, finiteness of weighted volume and polynomial volume growth for f-minimal submanifolds in a gradient shrinking soliton when f is a convex function. There are also others' works, for instance, see [6,9,15,17,22] and etc.…”