A barrier principle at infinity for varifolds with bounded mean curvature

Abstract

Our work investigates varifolds $\Sigma \subset M$ in a Riemannian manifold, with arbitrary codimension and bounded mean curvature, contained in an open domain $\Omega$. Under mild assumptions on the curvatures of $M$ and on $\partial \Omega$, also allowing for certain singularities of $\partial \Omega$, we prove a barrier principle at infinity, namely we show that the distance of $\Sigma$ to $\partial \Omega$ is attained on $\partial \Sigma$. Our theorem is a consequence of sharp maximum principles at infinity on varifolds, of independent interest.

arXiv comment38 pages, revised version. Some arguments have been clarified, typos corrected. Accepted in J. London Math. Soc

References in corpus (6)

• The mean curvature of cylindrically bounded submanifolds
• Weakly differentiable functions on varifolds
• Sectional and intermediate Ricci curvature lower bounds via Optimal Transport
• Duality between Ahlfors-Liouville and Khas'minskii properties for nonlinear equations
• On the essential spectrum of Nadirashvili-Martin-Morales minimal surfaces
• Minimal surfaces in minimally convex domains

Cited by in corpus (3)

• Stochastic half-space theorems for minimal surfaces and $H$-surfaces of ${\mathbb{R}}^3$
• Domains without parabolic minimal submanifolds and weakly hyperbolic domains
• On the principal eigenvalue of the truncated Laplacian, and submanifolds with bounded mean curvature