# Complete bounded $\lambda$-hypersurfaces in the weighted volume-preserving mean curvature flow

## TL;DR

This paper investigates the geometric and topological properties of complete bounded $mbda$-hypersurfaces evolving under weighted volume-preserving mean curvature flow, providing volume comparison, diameter estimates, and topological insights.

## Contribution

It introduces new volume comparison theorems, diameter estimates, and topological properties for $mbda$-hypersurfaces in the context of weighted volume-preserving mean curvature flow.

## Key findings

- Established volume comparison theorems for $mbda$-hypersurfaces with bounded second fundamental form.
- Derived estimates relating $mbda$, extrinsic radius, intrinsic diameter, and dimension.
- Identified topological properties of bounded $mbda$-hypersurfaces under natural restrictions.

## Abstract

In this paper, we study the complete bounded $\lambda$-hypersurfaces in weighted volume-preserving mean curvature flow. Firstly, we investigate the volume comparison theorem of complete bounded $\lambda$-hypersurfaces with $|A|\leq\alpha$ and get some applications of the volume comparison theorem. Secondly, we consider the relation among $\lambda$, extrinsic radius $k$, intrinsic diameter $d$, and dimension $n$ of the complete $\lambda$-hypersurface, and we obtain some estimates for the intrinsic diameter and the extrinsic radius. At last, we get some topological properties of the bounded $\lambda$-hypersurface with some natural and general restrictions.