New weighted geometric inequalities for hypersurfaces in space forms

Abstract

We prove a family of new sharp geometric inequalities involving weighted curvature integrals and quermassintegrals for smooth closed hypersurfaces in space forms.The tools we shall use are the inverse curvature flow by Gerhardt and Urbas and the locally constrained curvature flows introduced recently by Brendle, Guan and Li.M S C 2 0 2 0 53E10 (primary), 52A39 (secondary)

Weighted Alexandrov–Fenchel type inequalities for hypersurfaces in Rn$\mathbb {R}^n$

“…Inequality (1.3) was proved by Girão and Rodrigues [13] for 𝑘 = 1, 𝑙 = 0 and by Kwong and Miao [18] for general 𝑘 and 0 ⩽ 𝑙 ⩽ 𝑘 − 2. Based on Kwong and Miao's result [18], (1.3) was recently proved by Wei and Zhou [27] for general 𝑘 and 𝑙. We also presented another proof of (1.3) in [28] without using Kwong and Miao's results.…”

“…Based on Kwong and Miao's result [18], (1.3) was recently proved by Wei and Zhou [27] for general 𝑘 and 𝑙. We also presented another proof of (1.3) in [28] without using Kwong and Miao's results.…”

“…Theorem A [13,18,27]. Let Σ be a smooth, closed, star-shaped and 𝑘-convex hypersurface in ℝ 𝑛 (𝑛 ⩾ 3).…”