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In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space Hn+1 ${\mathbb{H}}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in Hn+1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as follows(0.1)∫Mλ′f2E12+|∇Mf|2−∫M∇̄fλ′,ν+∫∂Mf≥ωn1n∫Mfnn−1n−1n $$\underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{1}^{2}+\vert {\nabla }^{M}f{\vert }^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle +\underset{\partial M}{\int }f\ge {\omega }_{n}^{\frac{1}{n}}{\left(\underset{M}{\int }{f}^{\frac{n}{n-1}}\right)}^{\frac{n-1}{n}}$$ provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in (“A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold,” Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in Hn+1 ${\mathbb{H}}^{n+1}$ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in Hn+1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as below(0.2)∫Mλ′f2Ek2+|∇Mf|2Ek−12−∫M∇̄fλ′,ν⋅Ek−1+∫∂Mf⋅Ek−1≥pk◦q1−1(W1(Ω))1n−k+1∫Mfn−k+1n−k⋅Ek−1n−kn−k+1 \begin{align}\hfill & \underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{k}^{2}+\vert {\nabla }^{M}f{\vert }^{2}{E}_{k-1}^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle \cdot {E}_{k-1}+\underset{\partial M}{\int }f\cdot {E}_{k-1}\hfill \\ \hfill & \quad \ge {\left({p}_{k}{\circ}{q}_{1}^{-1}\left({W}_{1}\left({\Omega}\right)\right)\right)}^{\frac{1}{n-k+1}}{\left(\underset{M}{\int }{f}^{\frac{n-k+1}{n-k}}\cdot {E}_{k-1}\right)}^{\frac{n-k}{n-k+1}}\hfill \end{align} provided that M is h-convex and Ω is the domain enclosed by M, p k(r) = ω n(λ′)k−1, W1(Ω)=1n|M| ${W}_{1}\left({\Omega}\right)=\frac{1}{n}\vert M\vert $ , λ′(r) = coshr, q1(r)=W1Srn+1 ${q}_{1}\left(r\right)={W}_{1}\left({S}_{r}^{n+1}\right)$ , the area for a geodesic sphere of radius r, and q1−1 ${q}_{1}^{-1}$ is the inverse function of q 1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in (“Locally constrained curvature flows and geometric inequalities in hyperbolic space,” Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022).

Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C0-estimate technique under weaker conditions. As an application of this inverse Gauss curvature flow, the present paper first arrives at a non-even smooth solution to the Orlicz Minkowski problem.

In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1 2 + | ∇ M f | 2 − ∫ M ∇ ̄ f λ ′ , ν + ∫ ∂ M f ≥ ω n 1 n ∫ M f n n − 1 n − 1 n provided that M is h-convex and f is a positive smooth function, where λ′(r) = coshr. In particular, when f is of constant, (0.1) coincides with the Minkowski type inequality stated by Brendle, Hung, and Wang in ("A Minkowski inequality for hypersurfaces in the anti-de Sitter-Schwarzschild manifold," Commun. Pure Appl. Math., vol. 69, no. 1, pp. 124–144, 2016). Further, we also establish and confirm a new sharp Michael-Simon inequality for the kth mean curvatures in H n + 1 by virtue of the Brendle-Guan-Li's flow ("An inverse curvature type hypersurface flow in H n + 1 ," (Preprint)) as below (0.2) ∫ M λ ′ f 2 E k 2 + | ∇ M f | 2 E k − 1 2 − ∫ M ∇ ̄ f λ ′ , ν ⋅ E k − 1 + ∫ ∂ M f ⋅ E k − 1 ≥ p k ◦ q 1 − 1 ( W 1 (Ω)) 1 n − k + 1 ∫ M f n − k + 1 n − k ⋅ E k − 1 n − k n − k + 1 provided that M is h-convex and Ω is the domain enclosed by M, pk(r) = ωn(λ′)k−1, W 1 (Ω) = 1 n | M | , λ′(r) = coshr, q 1 (r) = W 1 S r n + 1 , the area for a geodesic sphere of radius r, and q 1 − 1 is the inverse function of q1. In particular, when f is of constant and k is odd, (0.2) is exactly the weighted Alexandrov–Fenchel inequalities proven by Hu, Li, and Wei in ("Locally constrained curvature flows and geometric inequalities in hyperbolic space," Math. Ann., vol. 382, nos. 3–4, pp. 1425–1474, 2022). [ABSTRACT FROM AUTHOR]