Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature

Ding, Qi

doi:10.2140/apde.2025.18.2537

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

Liouville theorem for minimal graphs over manifolds of nonnegative Ricci curvature

Qi Ding

Vol. 18 (2025), No. 10, 2537–2550

DOI: 10.2140/apde.2025.18.2537

Abstract

Let $Σ$ be a complete Riemannian manifold of nonnegative Ricci curvature. We prove a Liouville-type theorem: every smooth solution $u$ to the minimal hypersurface equation on $Σ$ is a constant provided $u$ has sublinear growth for its negative part. Here, the sublinear growth condition is sharp. Our proof relies on a gradient estimate for minimal graphs over $Σ$ with small linear growth of the negative parts of graphic functions via iteration.

Keywords

Liouville theorem, minimal hypersurface equation, nonnegative Ricci curvature, (sub)linear growth

Mathematical Subject Classification

Primary: 53A10

Milestones

Received: 25 March 2024

Accepted: 26 November 2024

Published: 4 November 2025

Authors

Qi Ding
Shanghai Center for Mathematical Sciences Fudan University Shanghai China

Open Access made possible by participating institutions via Subscribe to Open.

Vol. 18, No. 10, 2025