Constant weighted mean curvature hypersurfaces in Shrinking Ricci Solitons

TL;DR

This paper investigates the geometric properties of constant weighted mean curvature hypersurfaces within shrinking Ricci solitons, establishing conditions under which these hypersurfaces are level sets of the potential function or generalized cylinders.

Contribution

It provides new results characterizing when constant weighted mean curvature hypersurfaces are level sets or generalized cylinders in shrinking Ricci solitons, including Gaussian cases.

Findings

Hypersurfaces with finite weighted volume cannot lie in certain level set regions unless they are the level set.

Compact hypersurfaces with bounded mean curvature are level sets of the potential function.

In Gaussian shrinking Ricci solitons, certain integral conditions imply hypersurfaces are generalized cylinders.

Abstract

In this paper, we study constant weighted mean curvature hypersurfaces in shrinking Ricci solitons. First, we show that a constant weighted mean curvature hypersurface with finite weighted volume cannot lie in a region determined by a special level set of the potential function, unless it is the level set. Next, we show that a compact constant weighted mean curvature hypersurface with a certain upper bound or lower bound on the mean curvature is a level set of the potential function. We can apply both results to the cylinder shrinking Ricci soliton ambient space. Finally, we show that a constant weighted mean curvature hypersurface in the Gaussian shrinking Ricci soliton (not necessarily properly immersed) with a certain assumption on the integral of the second fundamental form must be a generalized cylinder.