Michael-Simon type inequalities in hyperbolic space Hn+1 ${\mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows

• Published in: Advanced nonlinear studies 2024-05, Vol.24 (3), p.720-733
• Main Authors: Cui Jingshi, Zhao Peibiao
• Format: Article
• Language: English
• Subjects: 35k96; kth mean curvatures; locally constrained curvature flow; michael-simon type inequality; primary 53e99; secondary 52a20
• ISSN: 2169-0375
• DOI: 10.1515/ans-2023-0127

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 constrai