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

TL;DR

This paper investigates the volume growth behavior of complete noncompact submanifolds in gradient Ricci solitons, establishing conditions under which they exhibit polynomial or linear volume growth, with implications for hypersurfaces in Euclidean space.

Contribution

It provides new results linking bounded weighted mean curvature to polynomial and linear volume growth in gradient Ricci solitons, extending understanding of submanifold geometry.

Findings

Submanifolds have polynomial volume growth under mild potential function conditions.

In bounded geometry, submanifolds exhibit at least linear volume growth.

Hypersurfaces in Euclidean space with bounded Gaussian weighted mean curvature have polynomial and linear volume growth.

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.