A Maximum Principle for hypersurfaces in $\mathbb{R}^{n+1}$ with an Ideal Contact at Infinity and Bounded Mean Curvature

TL;DR

This paper extends maximum principles for hypersurfaces in Euclidean space, including at infinity and with bounded mean curvature, generalizing previous results and addressing asymptotic behavior.

Contribution

It generalizes maximum principles at infinity and for hypersurfaces with bounded mean curvature, without Gaussian curvature restrictions, and extends Hopf's maximum principle asymptotically.

Findings

Established a maximum principle at infinity for hypersurfaces with bounded mean curvature.

Extended Hopf's maximum principle to hypersurfaces approaching asymptotically.

Provided new tools for analyzing hypersurfaces in Euclidean space with ideal contact at infinity.

Abstract

We will generalize a Maximum Principle at Infinity in the parabolic case given by De Lima [Ann. Global Anal. Geom. ${\bf 20}$, 325-343 2001] and De Lima and Meeks [Indiana Univ. Math. Journal ${\bf 53}$ 5, 1211-1223 2004], for disjoints hypersurfaces of $\mathbb{R}^{n+1}$ with bounded mean curvature without restrictions on the Gaussian curvature. We will also extend for hypersurfaces in $\mathbb{R}^{n+1}$ a generalization of Hopf's Maximum Principle for hypersurfaces that get close asymptotically.