Do Carmo's problem for CMC hypersurfaces in $\mathbb{R}^6$

TL;DR

The paper resolves Do Carmo's question in dimension $n=5$ by proving that complete noncompact constant mean curvature hypersurfaces in $\mathbb{R}^{6}$ with finite index must be minimal. The authors develop a strategy that combines a Sobolev-type inequality for finite-index CMC hypersurfaces, a finite-ends result, and a warped $\mu$-bubble construction to obtain uniform end-area/volume estimates, leading to a contradiction with noncompactness unless the hypersurface is minimal. The approach also yields alternative proofs for the cases $n=4$ and $n=3$, extending the reach of modern techniques in stable/finite-index CMC theory. Additional corollaries include a maximum principle at infinity and implications for min-max theory in low dimensions. The results rely on a careful blend of Sobolev, volume comparison, and variational bubble methods, adapted to the nonzero mean curvature setting.

Abstract

In this paper, we prove that complete noncompact constant mean curvature hypersurfaces in $\mathbb{R}^6$ with finite index must be minimal. This provides a positive answer to do Carmo's question in dimension $6$. The proof strategy is also applicable to $\mathbb{R}^4$ and $\mathbb{R}^5$, thereby providing alternative proofs for those previously resolved cases.

Theorems & Definitions (26)

• Theorem 1.1
• Corollary 1.2
• Theorem 2.1: Li-WangLi-Wang-finiteindex
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Theorem 2.5