Finite Index and Do Carmo Question for Constant Mean Curvature Hypersurfaces

TL;DR

The work investigates Do Carmo’s question for constant mean curvature hypersurfaces within the Euclidean setting under the finite $\delta$-index regime. It proves that subexponential volume growth precludes the existence of nonminimal, complete noncompact finite $\delta$-index $H$-hypersurfaces in $\mathbb{R}^{n+1}$, linking volume growth to spectral bounds via Brooks’ inequality. It further shows a rigidity result: if $\mathrm{Ric}_M \ge -\frac{4(1-\delta)}{n-1}|A|^2 g$, then any complete noncompact finite $\delta$-index $H$-hypersurface in $\mathbb{R}^{n+1}$ must be a hyperplane, established through a conformal change and criticality theory for $L_V=\Delta+V$ with $V=(1-\delta)|A|^2$. The paper develops a robust analytical framework, leveraging criticality theory to derive rigidity under geometric growth and curvature conditions, and extends the Do Carmo-type results to higher dimensions within the finite $\delta$-index class. Overall, it advances the understanding of when constant mean curvature hypersurfaces in Euclidean space must be minimal or trivial, using spectral and conformal methods.

Abstract

We prove that any finite $δ$-index hypersurface $M$ in ${\mathbb R}^{n+1}$ with constant mean curvature must be minimal, provided - the volume growth of $M$ is sub-exponential; - the Ricci curvature of $M$ satisfies $\operatorname{Ric}_M\geq -\frac{4(1-δ)}{n-1}|A|^2g,$ where $A$ is the second fundamental form and $g$ is the metric on $M.$ In the second case, our result further implies that, in addition to being minimal, such an $M$ must be a hyperplane. Notice that, we do not have any restriction on the dimension and that the second result is new also in the case of finite index hypersurfaces ($δ=0$).

Theorems & Definitions (38)

• Theorem 1
• Theorem 2
• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4: Brooks' Theorem
• Theorem 2.1
• proof
• Theorem 2.2
• Remark 2.3