Characterizations of Hyperplanes in Euclidean Space Through Higher-Order Anisotropic Mean Curvatures

Our aim in this article is to establish new characterizations of hyperplanes in Euclidean space via mild constraints on the higher-order anisotropic mean curvatures of an immersed complete hypersurface. Our approach is based on the study of the behavior of the height and angle functions naturally attached to a hypersurface, jointly with the use of appropriate maximum principles dealing with the notions of convergence at infinity and polynomial volume growth. In this context, we also obtain lower estimates for the minimum anisotropic relative null index of an immersed complete hypersurface, which can be regarded as a measure of how close such a hypersurface is to being flat.