Cui–Zhao's hyperbolic Michael–Simon inequality for higher mean curvatures

Let $M$ be a compact hypersurface in hyperbolic space $\mathbb{H}^{n+1}$, possibly with boundary $\partial M$, and let $f$ be a positive smooth function on $M$. Write $E_k:=E_k(\kappa)$ for the normalized $k$-th mean curvature, let $\nu$ be the unit outward normal of $M$, let $\bar{\nabla}$ be the Levi-Civita connection of the hyperbolic metric $\bar{g}$, and let $\lambda'$ denote the ambient radial weight used in the inequality. Cui–Zhao's conjecture. For $1\leq k\leq n$, there is a constant $C$ such that

$$\int_{M} \lambda' \sqrt{f^{2}E_{k}^{2}+|\nabla^{M}f|^{2}E_{k-1}^{2}}-\int_{M}\left\langle \bar{\nabla}(f\lambda'),\nu\right\rangle |E_{k-1}|+\int_{\partial M}f|E_{k-1}| \geq C\left(\int_{M}f^{\frac{n-k+1}{n-k}}|E_{k-1}|\right)^{\frac{n-k}{n-k+1}}.$$

When $k=1$, this becomes

$$\int_{M}\lambda'\sqrt{f^{2}E_{1}^{2}+|\nabla^{M}f|^{2}}-\int_{M}\left\langle \bar{\nabla}(f\lambda'),\nu\right\rangle+\int_{\partial M}f\geq C\left(\int_{M}f^{\frac{n}{n-1}}\right)^{\frac{n-1}{n}}.$$

This conjecture proposes a Michael–Simon type inequality adapted to hyperbolic space and higher mean curvatures, addressing the open problem of extending such inequalities from nonnegative to negative sectional curvature. The cited counterexample shows that the Euclidean-form inequality does not generally hold in negatively curved manifolds, motivating the additional hyperbolic terms.