Integral inequalities for closed linear Weingarten submanifolds in the product spaces

Fábio R Dos Santos; Sylvia F DA Silva; Antonio F DE Sousa

doi:10.1590/0001-3765202320230345

Integral inequalities for closed linear Weingarten submanifolds in the product spaces

An Acad Bras Cienc. 2023 Oct 30;95(3):e20230345. doi: 10.1590/0001-3765202320230345. eCollection 2023.

Authors

Fábio R Dos Santos¹, Sylvia F DA Silva¹, Antonio F DE Sousa¹

Affiliation

¹ Universidade Federal de Pernambuco, Departamento de Matemática, Av. Jornalista Anibal Fernandes, s/n, Cidade Universitária, 50740-540 Recife, PE, Brazil.

PMID: 37909571
DOI: 10.1590/0001-3765202320230345

Abstract

An integral inequality for closed linear Weingarten 𝑚-submanifolds with parallel normalized mean curvature vector field (pnmc lw-submanifolds) in the product spaces 𝑀𝑛(𝑐) × ℝ, 𝑛 > 𝑚 ≥ 4, where 𝑀𝑛(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 𝕊1(√1 - 𝑟2) × 𝕊𝑚-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.