Work

Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in ℝ⁴

Year: 2026

Type: article

Abstract: We describe a method to prove new integral inequalities for stable minimal hypersurfaces in Euclidean space. As an application, we give a simple proof that complete, two sided, stable minimal hypersurfaces in $\mathbb{R}^4$ are hyperplanes. A core part of the argument hinges on the fact that stable minimal hypersurfaces in non-negatively curved spaces are examples of manifolds with a spectral Ricci curvature lower bound; in particular, we prove a sharp pointwise gradient estimate for the Green kernel on non-parabolic manifolds with spectral Ricci lower bounds, extending previous work by Colding.

Source: Transactions of the American Mathematical Society

DOI: 10.1090/tran/9928

Authors: Xavier Cabré, Giovanni Catino, Luciano Mari, Paolo Mastrolia, Alberto Roncoroni

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: Ministero dell’Istruzione, dell’Università e della Ricerca, Universitat Politècnica de Catalunya

OpenAlex ID: W7220859965

FWCI: 21.98

Cites: 8

Cited by: 10

Related to: 10