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

TL;DR

The paper addresses radius bounds for nearly stable constant-mean-curvature ($H$) hypersurfaces in $(n+1)$-manifolds with a uniform lower bound on sectional curvature, for $n=2,3,4$. It extends classical radius estimates from stable surfaces to $\delta$-stable ones via a conformal change and the operator $L^{\delta}$, deriving intrinsic-distance bounds under explicit curvature and mean-curvature constraints, and establishing corollaries including scalar-curvature generalizations. It then proves a properness result: complete effectively embedded $H$-hypersurfaces with locally bounded second fundamental form are proper whenever $|H|$ exceeds a curvature-dependent threshold, leveraging the Stable Limit Leaf Theorem and nonexistence of stable limit leaves. Finally, it develops uniform graphical regularity for effectively embedded hypersurfaces with bounded $|A|$ in harmonic coordinates, extending classical graphical results to the effectively embedded setting. Together, these results generalize foundational properness and radius-estimate theorems (Colding–Minicozzi, Meeks–Rosenberg) to the realm of nearly stable CMC hypersurfaces in low dimensions, with consequences for embedding behavior and curvature-based corollaries.

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.

Theorems & Definitions (17)

• Theorem 1.1
• Theorem 1.2
• Definition 2.1
• proof : Proof of Theorem \ref{['radiusest']}.
• Corollary 2.1
• proof
• Theorem 2.2
• proof
• Corollary 2.3
• Definition 3.1