Radius estimates for nearly stable H-hypersurfaces of dimension 2, 3, and 4

TL;DR

This paper extends radius estimates for nearly stable constant mean curvature hypersurfaces in low-dimensional Riemannian manifolds with bounded sectional curvature, and proves properness for certain embedded hypersurfaces.

Contribution

It generalizes existing radius estimates to nearly stable CMC hypersurfaces in dimensions 2, 3, and 4, and establishes properness results for specific embedded hypersurfaces.

Findings

Radius estimates are extended to nearly stable CMC hypersurfaces in dimensions 2, 3, and 4.

Certain CMC hypersurfaces embedded in N are proven to be proper.

The work generalizes previous results by Rosenberg, Elbert, Nelli, and Cheng.

Abstract

In this paper we study the geometry of complete constant mean curvature (CMC) hypersurfaces immersed in an (n + 1)-dimensional Riemannian manifold N (n = 2, 3 and 4) with sectional curvatures uniformly bounded from below. We generalise radius estimates given by Rosenberg [32] (n = 2) and by Elbert, Nelli and Rosenberg [13] and Cheng [2] (n = 3, 4) to nearly stable CMC hypersurfaces immersed in N. We also prove that certain CMC hypersurfaces effectively embedded in N must be proper.