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.

Minimal submanifolds in a metric measure space

“…It is interesting to ask which character of hypersurfaces may let them at most polynomial volume growth. In [7], Vieira and we proved that a hypersurface in R nþ1 with bounded weighted mean curvature H f has at most polynomial volume growth. Moreover the following result was proved in [7] which generalised Theorem 3.3.…”

“…In [7], Vieira and we proved that a hypersurface in R nþ1 with bounded weighted mean curvature H f has at most polynomial volume growth. Moreover the following result was proved in [7] which generalised Theorem 3.3. Here, for convenience, we choose the normal constant 1 2 .…”

“…Very recently in order to study submanifolds immersed in a gradient shrinking soliton with weighted mean curvature vector bounded in norm, in [7], Vieira and we extended Theorem 3.1 to the case with Df À ahrf ; rf i þ bf a and Df a, where a [ 0; b [ 0 and a are constants (see [7, Theorem 1.1]). Then we obtained a result which generalizes Theorem 3.2, that is, for a properly immersed submanifolds R in a gradient shrinking soliton with weighted mean curvature vector bounded in norm, if the trace of Hessian of f restricted on its normal bundle satisfies tr R ?…”

“…r 2 f ! k 2 for some constant k, then it has polynomial volume growth and has finite weighted volume (see [7,Theorem 1.2]).…”

Volume Estimates and Classification Theorem for Constant Weighted Mean Curvature Hypersurfaces

“…Also, a consequence from one of their results is that for f = |x| 2 /4, sup − → H f , ∇f < ∞ and properness imply finite weighted volume and polynomial volume growth. Recently, Cheng, Vieira and Zhou [CVZ19] proved that for hypersurfaces in R n+1 with the norm of the weighted mean curvature bounded, properness is equivalent to polynomial volume growth. Here is our second result.…”