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The minimal surface condition, the vanishing of the trace of the second fundamental form, is wide enough to produce examples without positive curvature.  One condition leading to positive scalar curvature is that the cone over the minimal hypersurface has positive second variation proven by Schoen:  SchoenHyperconeMinimal.  In general, the formula $R = n(n-1)r = n(n-1) + n^2H^2 - S$ where $R$ is the scalar curvature and $S$ is the length squared of the second fundamental form holds from which one can deduce that for minimal surfaces $H=0$ the large size of $S$ will produce negative scalar curvature.  So $R \ge 0$ if and only if $S \le n(n-1)$ for the traceless second fundamental form length.  This may be the delicate issue for the mass gap problem in an S4 universe.