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A barrier principle at infinity for varifolds with bounded mean curvature

Eddygledson Souza Gama, Jorge H. S. de Lira, Luciano Mari, Adriano A. de Medeiros

Abstract

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

A barrier principle at infinity for varifolds with bounded mean curvature

Abstract

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

Paper Structure

This paper contains 7 sections, 11 theorems, 180 equations.

Table of Contents

  1. Introduction
  2. The smooth case
  3. Preliminaries
  4. Construction of the barrier
  5. Approximation
  6. Maximum principles at infinity for varifolds
  7. Proof of Theorem \ref{['omori-yau_vari']} and Corollary \ref{['cor_spectrum']}

Key Result

Theorem 1

Let $(M^m,\langle \, , \, \rangle)$ be a complete manifold satisfying for some $\ell \in \{2,\ldots, m-1\}$ and some $c \in \mathbb{R}$. Let $\Omega \subset M$ be an open set whose second fundamental form $\mathrm{II}_{\partial \Omega}$ in the inward direction satisfies in the barrier sense, for some constants $\Lambda_{\ell-1} \in \mathbb{R}$, $\Lambda_\ell \in [0, \infty)$, and that has locall

Theorems & Definitions (29)

  • Theorem 1
  • Remark 1: Regularity
  • Remark 2: On condition $\mathcal{P}_{\ell-1}^-[\mathrm{II}_{\partial \Omega}] \ge \Lambda_{\ell-1}$
  • Remark 3: On the locally bounded bending condition
  • Remark 4
  • Corollary 1
  • Definition 1
  • Remark 5
  • Lemma 1
  • proof
  • ...and 19 more