Volume estimates and classification theorem for constant weighted mean curvature hypersurfaces

TL;DR

This paper classifies complete embedded hypersurfaces with constant weighted mean curvature in Euclidean space, linking geometric properties to volume growth and second fundamental form characteristics.

Contribution

It provides a classification theorem for such hypersurfaces and establishes equivalences between properness, volume finiteness, and exponential growth.

Findings

Hyperplanes and round cylinders characterized by second fundamental form

Equivalence of properness, finite volume, and exponential growth for certain hypersurfaces

Classification results extend understanding of weighted mean curvature hypersurfaces

Abstract

In this paper, we prove a classification for complete embedded constant weighted mean curvature hypersurfaces $\Sigma\subset\mathbb{R}^{n+1}$. We characterize the hyperplanes and generalized round cylinders by using an intrinsic property on the norm of the second fundamental form. Furthermore, we prove an equivalence of properness, finite weighted volume and exponential volume growth for submanifolds with weighted mean curvature of at most linear growth.