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Michael-Simon type inequalities in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows
Advanced Nonlinear Studies ( IF 1.8 ) Pub Date : 2024-04-01 , DOI: 10.1515/ans-2023-0127 Jingshi Cui 1 , Peibiao Zhao 1
Advanced Nonlinear Studies ( IF 1.8 ) Pub Date : 2024-04-01 , DOI: 10.1515/ans-2023-0127 Jingshi Cui 1 , Peibiao Zhao 1
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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 H n + 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 H n + 1 ${\mathbb{H}}^{n+1}$ ,” (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 $$\underset{M}{\int }{\lambda }^{\prime }\sqrt{{f}^{2}{E}_{1}^{2}+\vert {\nabla }^{M}f{\vert }^{2}}-\underset{M}{\int }\langle \bar{\nabla }\left(f{\lambda }^{\prime }\right),\nu \rangle +\underset{\partial M}{\int }f\ge {\omega }_{n}^{\frac{1}{n}}{\left(\underset{M}{\int }{f}^{\frac{n}{n-1}}\right)}^{\frac{n-1}{n}}$$ 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 k th mean curvatures in H n + 1 ${\mathbb{H}}^{n+1}$ by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (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 \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 , W 1 ( Ω ) = 1 n | M | ${W}_{1}\left({\Omega}\right)=\frac{1}{n}\vert M\vert $ , λ ′(r ) = coshr , q 1 ( r ) = W 1 S r n + 1 ${q}_{1}\left(r\right)={W}_{1}\left({S}_{r}^{n+1}\right)$ , the area for a geodesic sphere of radius r , and q 1 − 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).
更新日期：2024-04-01
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