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 $(X^{ n+1},g)$.
Categories: math.DG