Stable Bernstein Problem in certain positively curved manifolds

TL;DR

This paper extends Bernstein-type theorems to certain positively curved manifolds, showing that stable minimal immersions are totally geodesic under specific curvature conditions, thereby generalizing previous results to higher dimensions.

Contribution

It proves new stable Bernstein theorems in higher-dimensional positively curved manifolds with specific curvature decay and positivity conditions, generalizing prior work.

Findings

Stable minimal immersions are totally geodesic under non-negative Ricci and BiRic curvature with decay.

No stable minimal immersions exist in manifolds with positive Ricci and non-negative (n-1)-Ricci curvature.

Results extend Bernstein theorems to higher dimensions with curvature assumptions.

Abstract

We formulate stable Bernstein type theorems in certain positively curved ambient manifolds. In all dimensions, we prove that for any complete Riemannian manifold $(X^{n+1},g)$, if the Ricci curvature is non-negative and it positive BiRic curvature with $\alpha$-decay, then any complete, two-sided, stable minimal immersion must be totally geodesic and $\text{Ric}(\nu,\nu)$ vanish along the minimal immersion. For $4\leq n+1\leq 6$, we prove that the result still holds if $(X^{n+1},g)$ has uniform positive $3$-intermediate curvature and non-negative $(n-1)$-Ricci curvature, which generalize Chodosh-Li-Stryker's result \cite{chodosh2024complete} for $n+1=4$ to higher dimensions. As an immediate corollary, we show that, in all dimensions, for a complete Riemannian manifold $(X^{n+1},g)$, if it has uniform positive Ricci curvature and non-negative $(n-1)$-Ricci curvature then there is no (not necessarily) complete, two-sided, stable minimal immersion in $(X^{n+1},g)$.