Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature

Abstract

In this article, we study properly immersed complete noncompact submanifolds in a complete shrinking gradient Ricci soliton with weighted mean curvature vector bounded in norm. We prove that such a submanifold must have polynomial volume growth under some mild assumption on the potential function. On the other hand, if the ambient manifold is of bounded geometry, we prove that such a submanifold must have at least linear volume growth. In particular, we show that a properly immersed complete noncompact hypersurface in the Euclidean space with bounded Gaussian weighted mean curvature must have polynomial volume growth and at least linear volume growth.

arXiv comment26 pages

References in corpus (2)

• Complete $\lambda$-hypersurfaces of weighted volume-preserving mean curvature flow
• Ricci-mean curvature flows in gradient shrinking Ricci solitons