Integral inequalities for closed linear Weingarten submanifolds in the product spaces

Abstract

Abstract An integral inequality for closed linear Weingarten ��-submanifolds with parallel normalized mean curvature vector field (pnmc lw-submanifolds) in the product spaces ����(��) × ℝ, �� > �� ≥ 4, where ����(��) is a space form of constant sectional curvature �� ∈ {−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 ��1(√1 − ��2) × ����−1(��) with 0 < �� < 1 when �� = 1. In the case where �� = −1, is obtained an integral inequality whose sharpness is attained only in the totally umbilical hypersurfaces. When �� = 2 and �� = 3, an integral inequality is also obtained with equality happening in the totally umbilical hypersurfaces.