Integral inequalities for closed linear Weingarten submanifolds in the product spaces

Abstract

An integral inequality for closed linear Weingarten 𝑚-submanifolds with parallel normalized mean curvature vector field (pnmc lw-submanifolds) in the product spaces 𝑀 𝑛 (𝑐) × ℝ, 𝑛 > 𝑚 ≥ 4, where 𝑀 𝑛 (𝑐) is a space form of constant sectional curvature 𝑐 ∈ {−1, 1}, is proved. As an application is shown that the sharpness in this inequality is attained in the totally umbilical hypersurfaces, and in a certain family of standard product of the form 𝕊 1 (√1 − 𝑟 2 ) × 𝕊 𝑚−1 (𝑟) with 0 < 𝑟 < 1 when 𝑐 = 1. In the case where 𝑐 = −1, is obtained an integral inequality whose sharpness is attained only in the totally umbilical hypersurfaces. When 𝑚 = 2 and 𝑚 = 3, an integral inequality is also obtained with equality happening in the totally umbilical hypersurfaces.