Volume estimate about shrinkers

2011/06/24 by Cheng, Xu, Zhou, Detang · 3 citations
#Differential Geometry (math.DG) #FOS: Mathematics

paper · doi:10.48550/arxiv.1106.4950

Abstract

We derive a precise estimate on the volume growth of the level set of a potential function on a complete noncompact Riemannian manifold. As applications, we obtain the volume growth rate of a complete noncompact self-shrinker and a gradient shrinking Ricci soliton. We also prove the equivalence of weighted volume finiteness, polynomial volume growth and properness of an immersed self-shrinker in Euclidean space.

Cited by

• The Rigidity Theorem of Legendrian self-shrinkers
• The Dirichlet problem for the minimal surface system on smooth domains
• An eigenvalue estimate for self-shrinkers in a Ricci shirinker

Related

• Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature
• The volume growth of complete gradient shrinking Ricci solitons
• Volume growth estimates for Ricci solitons and quasi-Einstein manifolds
• Lower volume growth estimates for Self-shrinkers of mean curvature flow
• Volume growth estimates of gradient Ricci solitons