On minimal graphs of sublinear growth over manifolds with non-negative Ricci curvature

Author:

Giulio Colombo, Luciano Mari, Marco Rigoli

Keyword:

Mathematics, Differential Geometry, Differential Geometry (math.DG), Analysis of PDEs (math.AP)

journal:

date:

2023-10-23 16:00:00

Abstract

We prove that entire solutions of the minimal hypersurface equation \[ \mathrm{div}\left(\frac{Du}{\sqrt{1+|Du|^2}}\right) = 0 \] on a complete manifold with $\mathrm{Ric} \ge 0$, whose negative part grows like $\mathcal{O}(r/\log r)$ ($r$ the distance from a fixed origin), are constant. This extends the Bernstein Theorem for entire positive minimal graphs established in recent years. The proof depends on a new technique to get gradient bounds by means of integral estimates, which does not require any further geometric assumption on $M$.

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