On stable minimal surfaces in manifolds of positive bi-Ricci curvatures
The authors introduce a new and interesting curvature invariant for a Riemannian manifold \(M^m\) as follows. Let \(u,v\in T_pM\) be orthonormal tangent vectors. Then the bi-Ricci curvature is defined to be \[ BRc(u,v)=\text{Ric}(u)+\text{Ric}(v)-K(u,v) \] where \(K(u,v)\) is the sectional curvature of the plane spanned by \(u\) and \(v\). Positive sectional curvature implies positive bi-Ricci curvature and an example is given that shows that the converse is not true. The main result of the paper under review is motivated by the work of \textit{R. Schoen} and \textit{S.-T. Yau} [Commun. Math. Phys. 90, 575-579 (1983; Zbl 0541.53054)] and is the following: Assume \(BRc\geq\kappa > 0\) and \(m\leq 5\). Then any closed stable minimal hypersurface \(S\) in \(M^m\) satisfies \[ \text{diam }S\leq \sqrt{c(m)} \frac{\pi}{\sqrt{\kappa}} \] where \(c(3)= 4/3\), \(c(4)=2\) and \(c(5)=4\). Moreover, let \(\Gamma_1\) and \(\Gamma_2\) be submanifolds with codimension 2. If \(\text{dist}(\Gamma_1,\Gamma_2)>\sqrt{c(m)} (\pi/\sqrt{\kappa}),\) then \(\Gamma_1\) and \(\Gamma_2\) cannot be connected by a stable minimal hypersurface.
scientific article; zbMATH DE number 949887
scientific article