Bernstein-Moser-type results for nonlocal minimal graphs

TL;DR

This paper extends classical Bernstein and Moser theorems to nonlocal minimal graphs, establishing flatness and affine properties under various boundedness conditions and constant mean curvature assumptions.

Contribution

It generalizes classical results to the nonlocal setting, providing new flatness and affine theorems for nonlocal minimal graphs with bounded derivatives and constant mean curvature.

Findings

Nonlocal minimal graphs with bounded derivatives are flat.

Graphs bounded on one side by a cone are affine.

Graphs with constant nonlocal mean curvature are minimal.

Abstract

We prove a flatness result for entire nonlocal minimal graphs having some partial derivatives bounded from either above or below. This result generalizes fractional versions of classical theorems due to Bernstein and Moser. Our arguments rely on a general splitting result for blow-downs of nonlocal minimal graphs. Employing similar ideas, we establish that entire nonlocal minimal graphs bounded on one side by a cone are affine. Moreover, we show that entire graphs having constant nonlocal mean curvature are minimal, thus extending a celebrated result of Chern on classical CMC graphs.