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arXiv · 1909.05724

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

Xu Cheng·Matheus Vieira·Detang Zhou
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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.

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Xu Cheng, Matheus Vieira, Detang Zhou. 2019-09-12. Volume growth of complete submanifolds in gradient Ricci Solitons with bounded weighted mean curvature. https://arxiv.org/abs/1909.05724

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