Michael-Simon type inequalities in hyperbolic space H n + 1 H^n+1 via Brendle-Guan-Li’s flows

Abstract 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 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 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 M ^ f^{2E₁^2+ ^Mf ^2}-M (f ^), + M f ₍^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) = cosh r. 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 H^n+1 by virtue of the Brendle-Guan-Li’s flow (“An inverse curvature type hypersurface flow in H n + 1 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 align & M ^ f^{2E₊^2+ ^Mf ^2E₊-₁^2}-M (f ^), E₊-₁+ M f E₊-₁ \\ & ({p₊q₁^-1 (W₁ () ) ) }^1{n-k+1} (M{ f^n-k+1{n-k} E₊-₁) }^n-k{n-k+1} 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 | W₁ () =1n M, λ ′ (r) = cosh r, q 1 (r) = W 1 S r n + 1 q₁ (r) =W₁ (Sₑ^n+1), the area for a geodesic sphere of radius r, and q 1 − 1 q₁^-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).