Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

Xavier Cabre; Giovanni Catino; Luciano Mari; Paolo Mastrolia; Alberto Roncoroni

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arXiv · 2604.14393

Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$

Xavier Cabre·Giovanni Catino·Luciano Mari·Paolo Mastrolia·Alberto Roncoroni

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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.

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Xavier Cabre, Giovanni Catino, Luciano Mari, Paolo Mastrolia, Alberto Roncoroni. 2026-04-15. Gradient estimates for the Green kernel under spectral Ricci bounds, and the stable Bernstein theorem in $\mathbb{R}^4$. https://arxiv.org/abs/2604.14393

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